Second Harmonic Alternating Current Polarography with a Reversible Electrode Process D. E. SMITH and W. H. REINMUTH Departmenf of Chemistry, Colombia University, New York 27, N. Y.
b A quantitative study of the second harmonic alternating current component arising in a.c. polarography with a reversible electrode process is presented. Results obtained with the model system, Fe(C~O&-~/Fe(C204)3-~, were found to b e in agreement with theoretical predictions of Tachi and Senda. The base current present in second harmonic measurements was found to b e extremely low. Advantages and applications of second harmonic a.c. polarography are discussed briefly. The circuit for the frequency selective amplifier used in this work is given.
Bauer (3) has discussed the analytical applications of this technique. Advantages in kinetic studies arise, not only from the reduction of the capacitive current, but also from the fact that higher harmonics are inherently more sensitive to variations in the kinetics and mechanisms of electrode processes. The present study, in addition to its primary purpose, indicates the efficiency with which the capacitive current can be eliminated by using second harmonic a.c. polarography.
T
By a general approach based on the assumption of linear diffusion, Tachi and Senda (18) derived equations for the current arising in a.c. polarography ITith a reversible electrode process. For an applied potential, V = Ed .. AEsin(wt) and nFAE/2RT < 1, the current is given by the series:
measurement of the second harmonic alternating current arising in a.c. polarography has been utilized by van Cakenberghe (6) and Bauer and Elving (4) for the determination of transfer coefficients. However, significant differences existed between the values of transfer coefficients determined by this method and by other means (4). Interest in possible causes for this inconsistency led to the consideration of second harmonic measurements. Rather than beginning with the case in which the current is controlled by the kinetics of the electrode process, as in transfer coefficient measurement, it was deemed advisable to study first a model system in which the Nernst equation is obeyed. Most theoretical treatments of a.c. polarography have neglected higher harmonics. However, Tachi and Senda (18) have presented equations for the second harmonic current with a reversible electrode process. To test the validity of this treatment, a careful quantitative study of a selected system, ferrous-ferric oxalate, was undertaken. Interest in the measurement of second harmonics is not based solely on its use in determining transfer coefficients. Since the double layer charging process represents an almost perfectly linear circuit element, whereas the faradaic process is somewhat nonlinear, contributions to higher harmonic a.c. components are predominantly faradaic. This reduction or elimination of the capacitive current by second harmonic measurement has definite advantages in chemical analysis and in the study of kinetics of electrode processes. HE
482
ANALYTICAL CHEMISTRY
THEORY
+
I
+ 1/; sin(wt + n/4) + &?A* ( 2 w t + s/4) + 2/13,sin (3wt + n/4) + . . .]f (1)
=
nFAjA0/2d*t
[I31
COS
where
(4)
When AE _< 10 mv., only the first terms are significant in Equations 3 and 4. In this case, after rearrangement and simplification, the expressions for the first and second harmonic a x . components are (when C: = 0):
sin (ut
+ n/4)
(10)
I(2wt) = & n 3 F 3 A C ~ ~ w ~ A E z1+$ (+$) 4R2T2 ( 1 +$)a sin (2ot - ~ / 4 ) ( 1 1 )
+
(12)
where C: = original concentration of oxidized Grm CS = original concentration of reduced
f'orm
Do = diffusion coefficient of oxidized D,
=
form diffusion coefficient of reduced form electrode size
A = and R, T , n, and F have their usual meaning.
At small amplitudes the first and second harmonic a.c. polarograms should be directly proportional to the first and second derivatives, respectively, of the conventional (d.c.) polarogram. The major assumptions utilized in this treatment are that small amplitudes are used and the laws of linear diffusion are obeyed. The use of small amplitudes is not a serious experimental limitation. Reasonably accurate work was carried out at amplitudes as low as AE = 3 mv. with the instrument used in this work. Rigorous mathematical arguments presented by Koutecky (14) and Gerischer (IO) have shown that, for the alternating current components, the treatment based on the assumption of linear diffusion is an excellent approximation for the dropping mercury electrode except for extremely small drops or very low frequencies. Specifically, for 2/2ZZ&w much less than the radius of the drop, the curvature of the drop may be neglected, and for drop time larger than the period of the alternating potential, convection due to drop expansion is negligible. Equation 10 for the first harmonic has been derived in various ways by several authors (5, 8, 17, 18). It was found to predict the experimental behavior accurately under the conditions assumed in the theory (6).
a harmonic component which, for hE 5 10 mv., is usually less than 5% of the total alternating current. The circuit selected for this purpose consisted of a twin-T filter network (9) in the feedback loop of a Philbrick K2-X operational amplifier. Two stages and maintenance of *I% tolerances for the resistors and condensers in the twinT networks were necessary to obtain the proper selectivity. Figure 1 gives the complete circuit diagram for this unit. Figure 1.
Frequency selective amplifier
C1 = C2 = Cl’ = C2’ = 0.50 pf. ca = cat = 1.00 pf. Resonant frequency control, varied through rotary R1, Rt, RB, RI’, Rz’, Ra’. switch and bank of resistors RB’ = R/2 R1 = Rz = R1’ = Rz’ = R:Rs Resonant frequency = 3.1 8 X 1 O S / R (with condensor sizes indicated) %, RE. 2-megohm pots. RB, R;. Gain control, varied through rotary switch and bank of resistors
A similar experimental test of Equation 11 for the second harmonic was the primary purpose of the work presented in this paper. EXPERIMENTAL
The basic electronic circuitry was similar to that described by Kelly, Fisher, and Jones (12)and DeFord ( 7 ) . In addition, a highly selective frequency filter was used to measure the second harmonic accurately in the presence of the much larger fundamental harmonic.
All polarograms were recorded on a Sargent Model MR recorder. The frequency and amplitude of the applied alternating potential were measured with a Hewlett-Packard Model 130B oscilloscope. The signal generator was a Hewlett-Packard Model 202A. A frequency filter was coupled with the signal generator to eliminate higher harmonics from the applied alternating potential. The d.c. potential scan rate was 17 mv. per minute. The rather slow scan rate was useful in obtaining the highest possible accuracy in the current-potential studies. The electrolysis cell was a Sargent Model S-29390. The solutions were thermostated at 30’ f 0.1”C. The capillary characteristics were: drop time, 6.47 seconds; mercury flow rate, 1.24 mg. per second. A mercury pool served as the reference electrode. The supporting electrolyte contained 0.30M potassium oxalate, 0.10X oxalic acid, and 0.0500X potassium chloride. The potassium chloride was added to stabilize the mercury pool potential. The p H of the solution was 3.72. Millimolar ferric ion was used in all zases except when testing the concentration dependence of the second harmonic current and the sensitivity of the instrument. All compounds were reagent grade. Calibration of the electronics was performed by replacing the cell with a standard resistor and measuring the recorder deflection resulting from an
alternating potential of known amplitude. By varying the size of the standard resistor and the amplitude of the alternating potential] a calibration curve of alternating current vs. recorder deflection was obtained. Calibration curves were linear except at low signal levels where diodes used for rectification introduced some nonlinearity. FREQUENCY SELECTIVE AMPLIFIER
Second harmonic a x . polarography requires a frequency selective circuit which will make it possible to measure
The 2-megohm variable resistors, Rd and RE, are necessary to prevent oscillation which occurs when the filter is tuned too finely. R4 and Rg are reduced until the oscillation is just eliminated. The K2-P amplifier and voltage divider stabilize the unit and eliminate d.c. offset. The operating procedure is simple. First, the resonant frequency of the amplifier is set to the desired value (see Figure 1). The frequency of the applied alternating potentiai is then carefully adjusted to match the resonant frequency of the amplifier. For second harmonic measurement, one must set the frequency of the applied potential to exactly one half the resonant frequency. At resonant frequency, gains of greater than 100 can be obtained up to 2000 C.P.S. Although the range of linear response decreased with increasing frequency, this presented no problem because the signal levels necessary for
0.ooJ $0.080
+0.040 (E,,cE,)
0.000
-0.040
-0.080
IN V O L T S j
Figure 2. Comparisons of calculated and observed second harmonic a.c. polarograms of ferrous-ferric oxalate system A.
Frequency = 38.4 c.p.s., AE Frequency = 77.0 c.p.r., A€ C. Frequency = 19.1 c.P.s., A € A, B, and C displaced vertlcally Calculated Observed
B.
-
= 5.00 mv. = 4.00 mv. = 5.00 mv. for clarity.
----
VOL. 33, NO. 4, APRIL 1961
483
-+a -z+-
0
--- A E - 4 . 0 0
--- CLE --- A E X --- A E
5 00
A
500
z
f---
W
AE
E
parisons of predicted and observed second harmonic polarograms. A total of 24 such comparisons were made a t amplitudes ( A E ) between 3 and 10 mv. and a t the frequencies mentioned. Differences between the predicted and observed values of the second harmonic current averaged 3% of the predicted value for -0.10 5 E d . o . -Eo 5 +0.10 volt. Differences between predicted and observed peak currents averaged about 2%. Comparisons of calculated and observed second harmonic currents were made a t 10-mv. intervals along each polarogram. The dependences of the second harmonic peak current on frequency and amplitude are given in Figures 3 and 4, respectively. The solid lines represent the theoretical values, while the points are the experimental values. Linear dependence on C," was also found in experiments for concentrations between 5 X 10-5M and 1 X 10-3M. The average deviation between predicted and observed currents a t a given concentration Tyas 2%. Essentially no difference was observed between the half-wave potential of the d.c. polarogram, the peak potential of the first harmonic a.c. polarogram, and the potential of zero
MV. 11
6.00
II
1.00
'I
8.00 "
0:
a
2
0.300.-
Y
U
W
0 K
I
n
z
0 0 W
2 00
(0
4
8 00
6 00
00
S Q U A R E R O O T OF F R E Q U E N C Y -3
Figure 3.
Second harmonic peak current vs. square root of frequency
recording were well within this linear range. I n practice, the selectivity of this unit was excellent. When measuring second harmonics no base current was observed with millimolar solutions of ferric ion up to 320 C.P.S. The limiting factor at low concentrations or low alternating potential amplitudes is electronic noise rather than frequency selectivity. I n some preliminary work testing the sensitivity of this instrument, no significant second harmonic base current was observed until concentrations were about 5 X 10-5M ferric ion. The limit of detection is around M for the ferric oxalate system and can be extended to about lO-'M for a 2electron reversible reduction. Because a major portion of the base current observed a t high sensitivities appeared to be electronic noise, which probably can be reduced, improvements on the instrument may extend the apparent limits by a factor of 10.
This fact and the further observation that both ferric and ferrous ions exist predominantly as trioxalato complexes in the medium employed (IS) indicate that the half-wave potential of the d.c. polarogram is equal to Eo for the couple. The second harmonic data were obtained a t 19.2, 38.4, and 77.0 C.P.S. The results of this study show very close agreement with Equation 11. Figure 2 contains representative com-
Table 1.
Potential Half-wave potential of d.c. POlarogram Peak potential of first harmonic a.c. polarogram Potential of zero second harmonic current
Values of Potentials
No. of
Result, Volt - 0.359 0.002
Conditions
Detns.
.....
2
-0.359 f 0.001
= 5 mv., varying frequency 19.1 c.p.s.,varying AE 38.6 c.P.s., varying AE 77.0c.p.s., varying AE
6
*
-0.357 i 0.003 -0.358 f 0.004 -0.357 f 0.003
aE
RESULTS AND DISCUSSION
A plot of the first harmonic peak current us. square root of frequency indicated that the ferrous-ferric oxalate system remained reversible up to a t least 320 C.P.S. The slope of the resulting straight line was used to determine Do,the only unknown parameter in Equations 10 and 11. The value of DO thus obtained was 0.494 X sq. cm. per second. Based on the Lingane-Loveridge equation for the diffusion current (16), this value of Do corresponds to a diffusion current conof 1.62 pa. per stant, Td/Co*m2/3t1/6, mmole per liter per mg. 2'3 seconds-1/2. This is in reasonable agreement with the value, 1.50, obtained by Lingane (15) in the presence of 0.005% gelatin. The difference may be attributable to the fact that no gelatin was used in the present work. The value of 4 is known to be 1.00 from conventional polarographic results presented by Kolthoff and Lingane (IS), which show that Do and D, are very nearly equal. 484
ANALYTICAL CHEMISTRY
0 - - -
6 0 . 0 .'
?
A--
/
19.2 C.P.S.
- 3 8 . 4 C.P.S.
I---
/''
/'
77.0 C.P.S.
20.0
AE'
Figure 4.
40.0
60.0
80.0
I N MV.' >-.
Second harmonic peak current vs. A€"
100.
8 7 7
second harmonic a t the frequencies used in this work. The values of these potentials us. the mercury pool are given in Table I. The results indicate that the derivation of Tachi and Senda (18) is valid and that Equation 11 accurately describes the behavior of the second harmonic current a t small amplitudes with a reversible electrode process. It is probable that addition of a second term (see Equation 4) to Equation 11 will extend the range of validity of this treatment to larger amplitudes. In addition to providing experimental verification for Equation 11, this work shows that, even under optimum conditions for conventional a x . polarography, second harmonic a.c. polarography can be performed with a t least the same degree of accuracy. At high frequencies or low concentrations the latter technique has decided advantages due to the very low base current.
Although the ultimate sensitivity of this technique in trace analysis will probably be no greater than squarewave and radio-frequency polarography (1, 2, 11), two of the most sensitive techniques a t present available, the much simpler electronic demands of second harmonic measurement are advantageous. LITERATURE CITED
(1) Barker, E G. C., Anal. Chim. Acta 18, 118 (1958). ((2) Barker, G. C., Jenkins, I. L., Analyst 77, 685 (1952). H. H., J . Electroanal. Chem. 1 , ((3) 3 7 Bauer, 2 256 (1960). (1980). (4) Bauer, H. H., Elving, P. J., ANAL. CHEW30, 341 (1958). 119581. (5) Breyer, B., B, Hacobian, S.,Australian Chen 7, 225 (1954). J . Chem. (6) Cakenberghe, Caker J. van, Bull. SOC. chim. belges 60, 337(1951). (7) DeFord, C D. D., Division of Analytical 133id Meeting, ACS; Chemistry, 133rd ACS, San Francisco, Calif., April 1958. ’
(8) Delahay, P.,
“New Instrumental Methods in Electrochemistry,” pp. 16872. Interscience. New York. 1954. (9) Fieisher, H., in M.I.T. Radiation Lab Series, G. E. Valley, Jr., H. Wallman, ede., Vol. 18, DD. 387-8. MoGraw-Hill, New York, 1948. (10) Gerischer, H., 2.physik. Chem. 198, 286 (19511. \ - - - - I
( l l ) k a r n m , R. E., ANAL.CHEM.30, 350
(1958). (12) Kelley, M. T., Fisher, D. J., Jones, H. C., Ibid., 31, 1475 (1959); 32, 1262 (1960). (13) Kolthoff, I. M., Lingane, J. J., “Polarography,” pp. 217-20, Interscience, New York, 1952. (14) Koutecky, J., Collection Czechoslov. Chem. Communs. 21, 433 (1956). (15) Lingane, J. J., J . Am. Chem. SOC.68, 2448 (1946). (16) Lingane, J. J., Loveridge, B. A, Zbid., 72, 438 (1950). (17) Matsuda, H., 2.Elektrochem. 61,489 i \ 16.57’1. - _ _,. .
(18) Tachi, I., Senda, M., BdZ. Chetn. SOC.Japan 28, 632 (1955).
RECEIVED for review October 27, 1960. Accepted January 11, 1961.
Distortion of Chronopotentiograms from Double Layer and Surface Roughness Effects W. H. REINMUTH Deparfmenf o f Chemistry, Columbia University, New York 27, N .
b Semiquantitative discussion is given of the effect of surface roughness and double layer capacitance on experimental chronopotentiometric potentialtime curves. Methods of minimizing and correcting for these effects are considered.
E
chronopotentiometric potential-time curves under certain conditions deviate markedly from the predictions of simple theory. These deviations are of practical interest because they are generally of such a nature as to increase the uncertainty of transition time measurement and invariably cause the measured times to shorn deviations from theory. At high current densities, corresponding t o short transition times, two major causes of distortion have been suggested: Charging of the capacitance of the electrical double layer may consume an appreciable fraction of the applied current ( 1 , 7 ) , and roughness of solid electrodes may cause the current. density to vary over the surface (1, 3). Empirical graphical methods have been proposed for the determination of transition times when such factors become important (4). However, it seemed of interest to inquire more carefully into the theory of these cases, in the hope that such inquiry might yield a better underXPERnimTAL
Y.
standing of the effects. Such is the aim of the present work. SURFACE ROUGHNESS
Exact characterization of a chronopotentiometric process a t a “rough” electrode requires some mathematical description of the geometry of the roughness. Even for such simple models as sinusoidal or sawtooth imperfections, rigorous solutions of the diffusion equations appear to be of forbidding complexity. However, even without such calculations, a number of observations can be made. If it is assumed that the oxidized and reduced forms of the electroactive couple have equal diffusion coefficients and that both are soluble in solution, it can be shown that the sum of their concentrations is a constant independent of time and position in the solution. This conclusion rests merely on the form of the diffusion equations and is independent of the geometry of the electrode. Moreover, if the Nernst equation is obeyed, the ratio of the concentrations a t any point on the electrode surface is the same, since the electrode, unless an unusually poor conductor of electricity, can be considered an equipotential surface. In conjunction, these two facts indicate that the concentra-
tion of the reactive species a t any instant is the same a t every point on the electrode surface. This conclusion is not significantly altered when the diffusion coefficients of the oxidized and reduced forms are unequal. The same treatment can be applied to cases of the deposition of insoluble species if the activity of the solid form can be assumed to be uniform over the electrode. Thus, the contention (2) that transition times are undefinable a t rough electrodes because the surface concentration becomes zero a t different times in various points on the surface does not appear to be generally true for reversible systems. This does not imply that the current density is also uniform over the electrode. It may vary markedly, tending to reach maxima a t points of convexity and minima a t points of concavity of the surface. However, it is of interest that high throwing power, an electroplater’s measure of uniformity of current density, is directly proportional to surface concentration of the reactive species. This suggests that the conclusion of the preceding paragraph is roughly applicable to irreversible systems as well as reversible ones. Although transition time thus appears t o be a definable quantity a t rough electrodes, the nonuniformity of current density indicates that the simplest VOL 33, NO. 4, APRIL 1961
485
