Symposium on Electroanalytical Instrumentation
ACS; Winter Meeting Analytical Division Phoenix, Ariz. Janaary 1966
Effect of Uncompensated Resistance on Electrode Kinetic and Adsorption Studies by Chronocoulometry GEORGE LAUER and ROBERT A. OSTERYOUNG North American Aviation Science Center, Thousand Oaks, Calif.
b The presence of “uncompensated resistance” in a potentiostatic experiment results in loss of potential control, which in turn affects the results of electrode kinetic and adsorption studies. A positive feedback potentiostatic circuit permits compensation of the potential drop across this uncompensated resistance. The circuit was tested by applying potential steps to an electrical analog of an electrodesolution interface and determining heterogeneous rate constants of Cd(ll) and TI(I) reduction by chronocoulometry at a hanging mercury drop electrode. The rate constants are essentially independent of the concentration of the electroactive ions; prior work without compensation had indicated an inverse relationship. The deliberate addition of uncompensated resistance results in a variation of the apparent rate constant, which is removed if the added resistance i s compensated by the electronic circuit. Problems in chronocoulometric studies of adsorption are ascribed to the presence of uncompensated resistance and an electrical analog of an interface where an adsorbed species reacts is proposed.
T
METHODS of potential step chronocoulometry and double potential step chronocoulometry have recently been shown to be useful techniques in the study of adsorption of electroactive species and of electrode reaction kinetics (1, 2, 6, 10). The initial studies of Christie] Lauer, and Osteryoung (6) reported the determination of the heterogeneous rate constants for Cd(I1) reduction using the chronocoulometric technique. Some variation of the rate constant obtained with concentration of Cd(I1) was indicated, but HE
1106
ANALYTICAL CHEMISTRY
ER = Et
Et
+
where r is the radius of the electrode, i is the net current, z is the distance of the reference probe from the surface of the electrode] K is the specific conductance of the solution, ER is the potential measured, Et is the true potential across the double layer. and R, is an “uncompensated” resistance “seen” by the reference electrode, which manifests itself in a potentiostatic circuit even though essentially no current flows through the electrode itself. The total “uncompensated” resistance is then the summation of the resistance due to the geometry of the electrode and that due to the capillary, the leads, etc. If we consider the standard analog of an ideally polarized electrode (Figure 1, a), when a potential step is applied at ER, the potential E t is given by
I (a1
Ru
(b)
Figure 1. a.
b.
Assumed electrical analog
Ideally polarizable electrode Polarized electrode with faradaic resistance
Rf
at that time it was not clear that the trend was not within experimental error. A more detailed examination of the procedure (8) established the dependence of the rate constant on concentration as a real effect outside the expected experimental error. In carrying out chronocoulometric studies on Tl(1) adsorption, it was recognized that the existence of uncompensated resistance, resulting in the loss of strict potential control, had a marked effect on the experimentally obtained charge curves (14). Some of the consequences of this situation have recently been discussed (12, IS), particularly as related to potential step experiments in which the current-time transient is analyzed. Nemec (11) has shown that the potential “seenJ1by a reference probe from a spherical microelectrode is given by
Et = E R ( ~
e-t’RC)
(2)
Thus the true potential, Et-that is, the potential across the double layer capacitor-is not equal to ER for a time determined by the RC time constant. If a faradaic current flows, the analog chosen is (Figure 1 , b) and ER never reaches Et,even asymptotically. Thus, there are two interconnected effects: the slow rise of the potential, and the difference in the final Et and the potential measured] ER. This problem is discussed quantitatively by aldham (12). The problem of ohmic drop can be overcome to a certain extent by positioning the reference capillary as close to the indicator electrode as possible. However, as Barnartt (3) has shown, the capillary will significantly alter the field if it is positioned too close to the surface.
I
1
I
RfGcdl 4
1
5 ma
2 ma
T
T
Figure 2. Schematic of potentiostatic circuit with compensation for ohmic losses
One horiz. div. equals 50psecs,
The problem becomes severe when spherical microelectrodes such as hanging mercury drops are used, since the radius is on the order of 0.5 mm. If the reference electrode is very far from the indicator electrode, then from Equation 1
The percentage reduction of R,(,, is then
7Figure 3. Current time traces following a potential step across an analog polarized electrode a. No compensotion b. Optimum compensotion c. Overcompensation
and amplifier 4 as a unity gain inverter. Considering only amplifier 1, the control amplifier,
eo Thus, it is necessary to position the reference electrode a t exactly one radius distance from the surface to obtain a 50y0reduction in R , and a t 0.111r for a 90% reduction. We have found it impractical to position a Luggin capillary tip reproducibly at such close distances. POSITIVE FEEDBACK SYSTEM
Instead of attempting to use a Luggin capillary, we have devised a positive feedback circuit which in essence compensates for the potential drop across R,, the ‘‘Uncompensated” resistance. [A recent paper by Pouli, Huff, and Pearson presents a compensation scheme which they indicate is operative under very limited conditions (26). Their analysis of the circuit appears to be in error, particularly in their concept of positive feedback and comments regarding reference electrode potentials.] The circuit is shown in Figure 2 .
A , is a Weking fast-rise potentiostat where the Sonde terminal corresponds to e2 and the Solspannung terminal corresponds to e,. The other amplifiers are manufactured by G. Philbrick Researches, Boston, Mass. A s is a Xodel P75AU and *A2 and A4 are Model SP656. The cell is represented by a resistor and a capacitor analog. Amplifier 3 is wired as a unity gain follower, amplifier 2 as an inverting current follower,
1
5 ma
Analog shown in p l a c e of cell, R, is resistance normolly compenroted by potentiortat
=
G(el
- ez)
(5)
where G is the open loop gain of the amplifier and eo, el, and e2 are voltages as indicated in Figure 2. If the input and feedback resistors are equal, we can write
E , - ez = e2 - E R (6) where E , is the negative of the potential desired, and ER is the potential “seen” by the reference electrode. All potentials are given with respect to ground. We define p as the fraction of the output of amplifier 4 fed back to the positive input of amplifier 1. Since the output of amplifier 4 is precisely i R , el = piR, (7) Solving Equation 6 for e2 and substituting together with Equation 7 into 5,we obtain
However, eo - i R , - iR, E t = E R - iR,, so that
2Et
=
Et and
+ 2iRu + 2iR, = G[2@iRm- ( E ,
+ Et + iRd1
(9)
If the potentiometer is set so that 0 = Ru/2R,, then - E , = E t , and the potentiostat controls the true potential across the double layer. If p > RU/2R,, the system will be out of control. Response to a potential step is a “worst case” test for an operational amplifier network. Ideally, the potential across the double layer should be identical to the input a t all times. In practice this is impossible. The initial response to a potential step requires an infinite current in a system with no uncompensated resistance. More realistically, for a 1-pf. capacitor to be charged in 1 microsecond, 1ampere average current is required. Operational amplifiers do not have infinite gain; the 6 db. per octave gain rolloff effectively limits the response. Thus, perfect compensation cannot be obtained using the step techniques if only because response of the amplifier system to the high frequency components of the step is degraded. It was therefore of interest to evaluate the actual system performance. POSITIVE FEEDBACK SYSTEM PERFORMANCE
The effective rise time of a potentiostatic system may be considered as that time required to charge the double layer. I n the case of an ideally polarized electrode, it is the time required for the current to go essentially to zero. As a suitable analog for an ideally polarizable electrode we used the circuit shown in Figure 2, where C d l is l l f . , R , = 150 ohms, R , is 100 ohms, and R , = m , In Figure 3 are shown the currenttime traces obtained for the above cirVOL. 38, NO. 9, AUGUST 1966
1107
ELECTRODE KINETIC STUDIES
1 7-
The variation of the heterogeneous rate constant was believed to result from the presence of uncompensated resistance. To investigate this hypothesis in some detail, the following experiments were carried out.
1 T
The heterogeneous rate constant,
2 ma
1ma
4 /--20,usec.
hh,for 5mM Tl(1) in 1F NaN08-
0.01F “ 0 , a t a hanging Hg drop was determined by chronocoulometry. I n this method the potential is stepped from a potential where no current flows to a potential along the polarographic wave. The charge, Q, passed following such a potential is recorded as a function of time, t . Only the species Ox is in solution, and the potential is stepped to a point where the reaction Ox ne + Red takes place. The charge-time behavior is analyzed by obtaining the asymptotic slope and intercept from a plot of Q us. tllz, since it has been shown (6) that
(b)
(a)
Figure 4. Current-time traces following a potential step using a hanging mercury drop (0.0291 sq. cm.) in 1 F NaNO3-0.01 M “ 0 3 a. b.
No compensation Optimum compensation
cuit with various values of 20Rm. In Figure 3, a, 2pR, = 0 and the rise time is greater than 500 microseconds. In Figure 3, b, 2pR, = 98 ohms and the rise time is approximately 80 microseconds. I n Figure 3, c, 2pRm = 110 ohms and “ringing” is observed even though 2pRn is less than 150 ohms. The ringing observed in Figure 3, c , is due to the deviations from ideality of the amplifier system. While the potential was kept constant 2pRm was increased; the system went into violent oscillation when 2pRn = 1 5 0 ohms. At this point the system is in complete positive feedback-Le., u n s t a b l e e v e n though no d.c. current is passing. This “break point” may be used to determine the uncompensated resistance, as discussed below. The system was then evaluated using a hanging mercury drop (0.0291 sq. cm.) in 1F NaN03 -0.OlM “0,. The i-t traces shown in Figure 4 were obtained with a Tektronix type 564 oscilloscope using a 3A72 dual trace amplifier and a 3B3 time base. Figure 4, a, is for 2pRm = 0. The theoretical current time behavior is given by
+
trace shown in Figure 4,a. The intercept log (AEIR) gives a value of R of 32 ohms, which is in agreement with the “break point’’ determination. The compensation procedure drastically decreased the effective nse time of the system.
where
i MA I
or AE l o g i = logR
1 t - 2.3 - RC
(12b)
Thus, with a known potential step, AE = 0.30 volt, a plot of log i us. t will have an intercept a t t = 0 of AE/R. Figure 4, b, shows the trace obtained when 2pR, was set so that the current barely showed no ringing (2pRm = 20 ohms). This value of p-i.e., just short of “ringing”-for a particular system is the maximum practical value of compensation used; the value of p a t this point will be designated by @*. The “break point” of the system occurred when 2@Rn = 32 ohms. Figure 5 shows the log i-t plot for the 1108
0
ANALYTICAL CHEMISTRY
.!
I
I
I
I
20
40
60
80
I
100
I
120
t p secs. Figure 5.
Log i vs. f plot for current decay shown in Figure 4, a
I1
0
K
=
nFAkatnCQx* exp
-anF(E - E") RT
Standard Rate Constant for 5 m M Tl(1) in 1 F NaNOs0.01F HNOa (Maximum value of is a*) Added resistance 0 ohm 50 ohms 100 ohms U B k8.h a P k8.h a B
Table 1.
ka,h
exp(1
- a) nF(ERT- E")\
0.55 0.40 0.31 0.23
(15)
is the apparent heterogeneous rate constant (cm. per second), n, F , A , and D have their usual meaning, and Cox* is the bulk concentration of species Ox (moles per cc.). A correction for the coulombs required to charge the double layer is made by measuring a AQ corresponding to a given step AE in the absence of electroactive species; AQ is then subtracted from the experimentally obtained Q-t curve. Resistance was deliberately added in series with the indicator electrode in order to evaluate the system. The relevant data obtained are shown in Table I. All the data were collected using a high-speed digital data conversion and acquisition system (9, 15). The apparent rate constant a t the operational maximum compensation, @*, is remarkably constant. The value of decreases with resistance when 2pRm = 0. The values of k a , h at intermediate values of 2@Rmshow less variation with added resistance as compensation is increased. If the ratio of 2P*Rm/ R , is a function of the electronic system
0.62 0.62 0.53 0.58
0.125 0.090 0.062 0
0.55 0.60 0.55 0.53
0.54 0.19 0.14 0.11
0.290 0.145 0.072 0
0.50 0.13 0.084 0.065
0.59 0.53 0.52 0.49
0.460 0.230 0.115 0
k8,h
I
0
20
10
0
compensated uncompensated
kS3h(comp)= 0 085 cmlsec
a (camp)= 0304
-k,.h (uncomp)=0.0371cmlsec
a (uncomp)=O 264
0
I
-590
I
-610
I
I
-630 -650 MV vs S.C.E.
1
-670
/I
5bO
-6
Figure 7. Log h vs. E plot for 1 OmM Cd(ll) in 1 F N a ~ S 0 4 Upper line corresponds to values obtained for
3
= By, lower for P = 0
/-
= compensated =uncompensated
r-/ I
IO1 -570
40
30
fl Rm ohms Figure 6 , Plot of OR,,, vs. added resistance at /3*
o
/
-610
-230
-6150
-6;O
-E
MV v s S.C.E.
Figure 8.
Log K vs. E plot for 1 OmM Cd(ll) in 1 F Na2SOI
Value of K at El/, is nFAC k r , h Upper line corresponds to values obtained for f3 =
By, lower for
VOL. 38, NO. 9, AUGUST 1966
@ = 0
1109
Ru I
1
Figure 10. Operational analog for electrode showing adsorption
IOpc
f
This analog is valid only for a potential step procedure. Potential true and corresponding current
-I Figure 9.
;I-
Q-f curves for potential step in
lOmM Cd(ll) in
1 F Na2S04 Potential step to a. j3=0 b. j3 = j3*
only, a constant percentage of R, should be compensated. Therefore, a plot of 2p*R, us. added resistance should be linear and the intercept should be the value of the uncompensated resistance of the cell alone. Figure 6 shows such a plot for the data presented in Table I. The value of R, from the intercept is 37 ohms, in reasonable agreement with the value (32 ohms) obtained for the same solution without electroactive species. The apparent heterogeneous rate constants for 5 and 10mM Cd(I1) in 1F Na2S04a t a hanging mercury drop were then determined and found to be 0.74 and 0.83 cm. per second, respectively. These numbers can be compared to those reported previously (6) and those reported by Lingane and Christie (8) using a similar method. In both previous cases an inverse relation was observed between k a , h and concentration. Plots of log us. E and log K us. E , with and without compensation, are shown in Figures 7 and 8 for 10mM Cd(I1). If one applies a sufficiently large step to the electrode, so that for t > 0, the surface concentration of the species undergoing reaction will be equal to 0, then the Q-t behavior will follow the chronoamperometric current integral 2nFACD,,1/2t1/2 Q =
&2
(16)
If, however, the surface concentration does not go to zero instantly, because of slow-rise equipment or uncompensated resistance, the Q a t t will be less than the value calculated from Equation 16. One may thus consider these coulombs ‘‘lost.” The error can be estimated in terms of a linear potential sweep followed by a steady potential as suggested by Booman and Holbrook (4). The charge passed a t any time following the application of a potential step can approach that given by Equation 16 11 10
ANALYTICAL CHEMISTRY
E1/2
only if the potential across the double layer is constant. If, as is the case, the actual potential rises slowly, the electron transfer rate will be determined by the actual potential during the rise, and thus less than the theoretical charge will have passed a t any time, t ; this will alter the concentration profile a t the electrode surface. Although this effect will be present in any potentiostatic method, chronocoulometry is most sensitive to this disturbance, since the integral of the current contains the history of the reaction from the time of initiation. The effect is accentuated by a high concentration of electroactive species and for potential steps to potentials around the polarographic half wave where the current is most sensitive to the true potential. Figure 9 shows Q-t traces obtained for lomill Cd(I1) compensated and uncompensated; the “lossJJof coulombs in the uncompensated case is striking. ARTIFACTS IN ADSORPTION STUDIES
In the study of adsorption of electroactive species by chronocoulometric techniques the potential is stepped from El where no faradaic current flows to a potential, EP,where the electron transfer rate is essentially sofast that the reaction is diffusion-controlled a t all times. In such a case it is assumed that the species adsorbed at El will reduce a t E, instantaneously. The problem of uncompensated resistance can be very serious in such cases, as pointed out by Anson, Christie, and Osteryoung, and can result in plateaus in the i-t traces (2). A reasonable analog for an electrode a t which an electroactive substance adsorbs a t El and then reduces, is shown in Figure 10, where cdl is the analog for the double layer capacity, D1 is a zener diode whose breakover potential is fairly low, and R, is the faradaic resistance. We consider the potential across the
double layer capacity for the circuit shown in Figure 10, following a potential step AE = E,-EI applied at R,; the potential will rise rapidly a t a rate determined by R,Cdr until the breakover potential of the zener diode is reached. At that point the potential rise will be significantly decreased because of charging of the capacitor in series with D1, which may be considered analogous to discharge of the adsorbed species. [This is not the analog considered by Laitinen and Randles in their studies of adsorption by means of faradaic impedance (‘7). The chronocoulometric technique can determine only the amount of material absorbed at El, the potential from which the step is applied, and a t which no net faradaic current flows. The faradaic impedance procedure considers the adsorption as a parallel reaction path to explain “anomalous” results. Thus, in general, the faradaic impedance considers the reaction in the vicinity of the polarographic half-wave potential. Recent considerations of the merits of the two techniques as applied to detecting adsorption by Laitinen and Chambers (6) and Anson (1) appear to have overlooked this point.] Figure 11, a, shows the potential rise across Cdl for a step from 1.0 to 3.5 volts for an analog system made up of R, = 100 ohms, R, = a, C d l = 1 pf., D1 = 1N702 (2.6 volts), C = 10 pf. The corresponding current is shown in Figure 11, b. Figure 11, b, shows that there is a distinct “plateau” in the current immediately following the initial current spike. However, if 60% of the resistance is compensated, one obtains the potential rise and corresponding current shown in Figure 12, a and b. The plateau has essentially disappeared and the potential rise across c d l has been sharpened. In a recent paper, Anson, Christie, and Osteryoung (2) showed i-2 traces for Cd(I1) in SCN- solutions where distinct plateaus were observed. The plateaus were assumed to occur because of reduction of the adsorbed species. Such interpretation is correct, however, only because of the presence of uncompensated resistance. A 2mM Cd(I1) solution in 0.5M NaSCN-0.5M NaW03 solution was run. Figure 13, u, shows the i-t trace where no compensation is applied;
u)
a
.--E
-
_t
E
.5 volts
f
z
f
1
L
Z
2.5
f 2 0 0 p5
H
t
, microseconds
a. w
-s
..E
'1
!t
L
0 ' 5 Figure 1 1.
Potential rise and corresponding current
True potential across C d l following application of potential step at Ei using analog shown in Figure 10 with p = 0 b. Corresponding i-1 trace
a.
f 450c.)-
1 , microseconds
(bf
FigurL 13. Current-time trace following a potential step in 2mM Cd(ll) in 0.5M NaSCN-O.5M N a N 0 3 a.
b.
.5 volts
p = O p = p*
Figure 13, b, shows the corresponding trace when compensation is applied. However, such plateaus where no compensation is applied are not conclusive evidence of adsorption. If the concentration of electroactive species is made sufficiently great, a plateau will be observed even in the absence of adsorption, the plateau current being due to a massive iR drop.
f
CONCLUSIONS
i IO m a
f Figure 12. Conditions identical to those in Figure 1 1, with fl = p* a. b.
True potential across c d l Corresponding i-1 trace
The ohmic resistance can be ccmpensated electronically to a degree dependent on the quality of the electronic circuitry. I n the case of potential step methods the scheme is effective and necessary. The compensation can be increased to higher values for voltage sweep methods than for the step method, as phase shift problems will be less severe. Similar considerations hold for a.c. polarographic methods ( I T ) . The method is effective for all potential control methods except polarography. This special problem will be treated in a separate communication. VOL. 38, NO. 9, AUGUST 1 9 6 6
1111
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
