Second Harmonic Alternating Current Polarography: Some Experimental Observations with Quasi-Reversible Processes Thomas G. McCord1 and Donald E. Smith2 Department of Chemistry, Northwestern University, Evanston, Ill. 60201 Results are presented of experimental measurements of the second harmonic ac polarographic response with the systems Cr(CN)68-/Cr(CN)64- in 1M KCN and Cd2+/ Cd(Hg) in 1M Na2S04. The data encompass an applied alternating potential frequency range of 23 Hz to 4.4 KHz. Agreement with theoretical predictions for the second harmonic ac polarographic response of quasireversible systems i s excellent. The best fit of theory with experimental data was obtained for k, = 0.25 crn sec-1 and ~r = 0.55 in the case of the Cr(CN)68-/Cr(CN)84- redox system. The corresponding values for the cadmium system were 0.15 cm sec-l and 0.30, respectively. In both cases, these rate parameter values are essentially identical to those obtained by fundamental harmonic measurements.
NUMEROUS WORKERS have considered the use of second harmonic ac polarographic measurements for electrochemical kinetic investigations (I-8)-see references (7) and (8) for a more complete bibliography. The attendant advantages of relatively low double-layer charging current contributions and notable sensitivity to kinetic effects are well documented. Despite this not insignificant activity encompassing a period exceeding 1 decades, the development of second harmonic ac polarography for kinetic applications must be considered in its infancy. Only the simple electrode reaction mechanism, O+ne$R
(R1)
has received detailed theoretical scrutiny (2, 4-8). Experimental studies have been primarily limited to systems allegedly following this simple mechanism (7, 8) and those directed to quantitatively testing the theory and obtaining kinetic parameters for the fundamentally important quasi-reversible case (mixed rate control by diffusion and heterogeneous charge transfer) are few (2, 4, 6). Recent instrumental developments (9-12) which greatly re1 NIH Graduate Fellow: present address, General Electric Corp., Materials and Processes Laboratory, Schenectady, N. Y.
12305 2
To whom correspondence should be addressed.
(1) K. B. Oldham, J. Electrochem. SOC.,107,766 (1960). (2) J. Paynter, Ph.D. Thesis, Columbia University, New York, 1964. (3) H. H. Bauer, J. Sci. I i d Res., 24, 372 (1965). (4) H. H. Bauer and D. C. S . Foo, Austr. J . Chem., 19, 1103 (1966). ( 5 ) T. G. McCord and D. E. Smith, ANAL.CHEM., 40,289 (1968). (6) J. E. B. Randles and D. R. Whitehouse, Trans. Furuduy SOC., 64, 1376 (1968). (7) B. Breyer and H. H. Bauer in “Chemical Analysis,” P. J. Elving and I. M. Kolthoff, Eds., Vol. 13, Interscience, New York, N. Y., 1963, Chap. 2. (8) D. E. Smith in “Electroanalytical Chemistry,” A. J. Bard, Ed., Vol. 1, M. Dekker, Inc., New York, N. Y., 1966, Chap. 1. (9) G. Lauer and R. A. Osteryoung, ANAL.CHEM., 38,1106 (1966). (10) E. R. Brown, T. G. McCord, D. E. Smith, and D. D. DeFord, ibid., p 1119. (11) E. R. Brown, D. E. Smith, and G. L. Booman, ibid., 40, 1411 (1968). (12) E. R. Brown, H. L. Hung, T. G. McCord, D. E. Smith, and G. L. Booman, ibid., p 1424.
duce the most serious problems associated with second harmonic measurements have accentuated the profound potentialities of this technique as has the availability of digital computer aids to cope with the cumbersome theoretical formulations. Aided by these developments, we have undertaken quantitative experimental measurements of the second harmonic response with a variety of systems to obtain more extensive experimental evidence regarding the validity of theoretical concepts and suggested advantages of second harmonic ac polarography. The present discussion is concerned with our observations of the second harmonic response with quasi-reversible systems under conventional ac polarographic conditions (with dc polarization). A detailed comparison of experimental results with theoretical predictions for the second harmonic wave shape and magnitude as a function of frequency is provided for the systems, Cr(CN)63-/Cr(CN)64in 1M KCN and Cd2+/Cd(Hg) in 1M Na2S04. The data presented exhibit outstanding agreement with theory validating, at least for the systems studied, the fundamental assumptions underlying the theoretical development for the quasireversible case. Particularly relevant to the present work is a recent paper by Randles and Whitehouse (6) describing excellent agreement between theory and experiment for detailed aspects of the second harmonic response with the V3+/V2+system under conditions of the classical faradaic impedance experimentLe., in absence of dc polarization. The data presented below add to the findings of Randles and Whitehouse and the less detailed earlier investigations (2, 4 ) in two ways. First, they reveal additional systems which behave in accord with theoretical predictions for the quasi-reversible case. Second, and perhaps more important, they establish that effects of dc polarization, which can have a crucial influence on the second harmonic response under ac polarographic conditions (5, 8 , 13), can be handled theoretically in an adequate manner. This is particularly relevant in the case of metal ion-metal amalgam systems where substantial spherical diffusion effects originate in the dc process (13-17). THEORY The most general theoretical formulation of the second harmonic ac polarographic wave presented to date for the quasi-reversible case has the form (5) Z(2wr) = Z ( 2 w ) W ( w ) sin (2wt Z(2w) =
+
+2)
n 2F2ACo*(2w D o ) 2AE 4RT cosh2
(3
(13) T. G. McCord, E. R. Brown, and D. E. Smith, ANAL.CHEM., 38, 1615 (1966). (14) T. Biegler and H. A. Laitinen, ibid., 37, 572 (1965). (15) J. R. Delmastro and D. E. Smith, J . Electroanal. Chem., 9, 192 (1965). (16) J. R. Delmastro and D. E. Smith, ANAL.CHEM.,38, 169 (1966). (17) Zbid., 39, 1050 (1967). VOL. 41, NO. 1, JANUARY 1969
131
The foregoing theoretical formulation was applied essentially unchanged to the analysis of data obtained with the Cr(CN)63-/Cr(CN)64-system. However, a significant modification of the above theory was invoked before analysis of the Cd2+/Cd(Hg)data. Studies of the influence of spherical diffusion on the fundamental and second harmonic ac polarographic waves (13-17) have shown that, although negligible for systems such as Cr(cN)~~-/cr(CN)6~-, the contribution of spherical diffusion with metal ion-metal amalgam systems like CdZ+/Cd(Hg)is invariably significant. Thus, the neglect of spherical diffusion in the above theoretical expressions is expected to lead to difficulties in the case of the cadmium system. However, the need to account for spherical diffusion effects is readily accommodated. Theoretical considerations based on the stationary sphere electrode model (13, 16) have indicated that, for diffusion-controlled dc polarization as one finds with the cadmium system, the spherical diffusion effect on all ac polarographic current components can be approximately accounted for by multiplying the current amplitude by a "spherical correction factor," F,(t). For metal ion-metal amalgam systems F,(t) is given by (16)
F&) L
+ (1 +
y2 -
=
1
(Do1"+ DR1l2) [l - exp (b2t)erfc (bt1i2)] + (ergo"2 - DR1j2)
where
Accordingly, for the analysis of the cadmium system, Equation 2 was modified to Z(2w)
=
n2F2ACo*(2w Do)l l 2AE
4RT cosh2
nF RT
j = - (Ed,
EllZr= Eo
- RT -In nF
- El/;)
(; ) (E?)"' ~
f = foyRa
(4)
(T)
F,(t)F(Xt1/2)G
2w"2
(19)
\ L /
Experimental assessment of the accuracy of F,(r) in accounting for spherical diffusion effects with the DME (13, 17) has indicated that it is slightly inaccurate, probably because of neglect of drop growth and shielding effects in its derivation. This apparent residual error was minimized in the present work by a procedure described in the data treatment section.
D = DoBDRa p = 1 - a A
=
0.8515 X 10-3(mt)2/s
Notation definitions are given below. The foregoing equations are based on the expanding plane model of the dropping mercury electrode (DME). The general nature of their predictions has been surveyed (5).
EXPERIMENTAL
Second harmonic measurements were effected with the aid of a potentiostat which allowed complete compensation of ohmic potential losses. Conditioning of the output (current) signal of the potentiostat amounted to tuned amplification ( 2 stages tuned to second harmonic), conventional full-wave rectification and filtering the rectifier output with a low-pass filter of Butterworth response. The resulting signal, rep-
b Figure 1. Second harmonic ac polarographic results for C I ~ C N ) ~ ~ - / C ~ ( Csystem N)~~System: 2.00 X 10-3MCr(CN)sa-in 1.00MKCN Applied: dc scan rate 10.0 mV per minute, 20.0 mV peak-to-peak sine wave: frequencies listed below Measured: second harmonic faradaic component at end of mechanically controlled drop life. Readout aided by sample-and-hold circuitry (10). Theoretical second harmonic polarogram for a = 0.55, k , = 0.25 cm sec-', D o = DR = 8.20 X lop6cm2sec-l, t = 4.93 sec, A = 0.0192 cm2,T = 25.0 "C, AE = 10.0 mV.,n = 1,C,* = 2.00 X 10-3M, and appropriatew value Observed second harmonic currents
-
A . Applied frequency = 23.0 Hz B. Applied frequency = 46.0 Hz C. Applied frequency = 83.0 Hz D . Applied frequency = 166.0 Hz
132
ANALYTICAL CHEMISTRY
E . Applied frequency = 332.0 Hz F. Applied frequency = 555.0 Hz G . Applied frequency = 1110 Hz H. Applied frequency = 4440 Hz
VOL. 41, NO. 1, JANUARY 1969
133
Edc I VOLTS
4.1
134
0
ANALYTICAL CHEMISTRY
vs SCE I
VOLTS VS ECE I
E*lMLTs
vs % E l
Ed. I VOLTS VS SCE
I
resenting the second harmonic current amplitude, was recorded as a function of the applied dc potential, utilizing sample-and-hold circuitry (IO) to effect current measurement at the end of the mechanically-controlled mercury drop life. By virtue of the elimination of iR drop effects and the negligibility of the second harmonic charging current under the conditions employed, the readout was taken as the faradaic second harmonic component unencumbered by nonfaradaic influences and was compared directly to theory. The potentiostat, all other aspects of the electrochemical instrument, and all peripheral supporting equipment (constant temperature baths, cells, etc.) employed in these measurements have been described in detail elsewhere (8, 10-12), as was the approach to calibrating the instrument response to second harmonic signals (18). Methods of compound and solution preparation, purification of nitrogen for polaropurification also graphic cell degassing and solvent (HzO) have been described earlier (IO,12, 17). Measurements were performed at 25.0 5 0.1 “C. The polarographic cell consisted of a DME working electrode (Sargent S-29417 capillary), a platinum wire auxiliary electrode, and a saturated calomel reference electrode. DATA TREATMENT
A Control Data Corporation Model 6400 digital computer aided by a Calcomp Model 565 digital incremental plotter for analog readout of theoretical polarograms enabled convenient assessment of the k , and a values which yielded the best match of theory and experiment. A preliminary estimate of the rate parameters was effected through the use of theoretical working curves involving peak potential measurements ( 5 ) which do not involve comparison of theory and experiment on the basis of absolute current magnitudes. Employing these preliminary k , and a values, a detailed comparison with theory of all aspects of the observed second harmonic response was carried out which allowed final refinement of the rate parameter values through trial-and-error fitting of experimental data to theory. The latter operation employed an apparent electrode area calculated as described below and known values of all other parameters, including diffusion coefficient values from the literature (19-21). Although not incorporated in the FORTRAN program employed for this purpose, the simplification F(Xt”2) = 1
(20)
was validated for both systems studied because the heterogeneous charge transfer rates were sufficiently rapid that the dc polarization processes were strictly diffusion-controlled ( 5 )*
Rather than calculate the electrode area from capillary characteristics and the appropriate relationship (Equation 16), (18) D. E. Smith and W. H. Reinmuth, ANAL.CHEM.,33, 482
(1961). (19) P. J. Lingane and J. Christie, J. Electroanal. Chem., 10, 284 (1965).
(20) D. E. Smith, Ph.D. Thesis, Columbia University, New York, N. Y., 1961. (21) B. Breyer and H. H. Bauer, Austr. J . Chem., 9,425 (1966).
4
which introduces a slight error due to neglect of the inactive area at the orifice of the capillary, an apparent electrode area was calculated empirically. The second harmonic anodic peak current at one particular frequency was the observable employed. With the aid of the preliminary estimates of k , and a , the values of the other parameters used in calculating k , and a , and the above theory, the electrode area value necessary for agreement of this particular experimental point with theory was calculated and taken as the apparent electrode area. If subsequent refinement of k , and CY (see above) caused a change in the theoretical value of the second harmonic peak current in question, the apparent electrode area was recalculated on the basis of the refined k , and a values. This was followed by recalculation of k , and a , if necessary, etc. This iterative procedure amounted to force-fitting experiment and theory at a single point so that the comparison between theory and experiment is not strictly absolute. For example, any constant determinant errors (e.g., an error in the Dovalue employed) would be compensated. In the case of the cadmium system, the apparent electrode area undoubtedly contains a correction for the error inherent in the theoretical spherical diffusion factor, F,(t). However, since this correction is obtained through a measurement at just one combination of Edc and w , the theoretical expression for F8(i) still must be accurate in its prediction of the spherical diffusion effect’s relative variation with Edc and lack of frequency dependence in order for theory and experiment to agree at other values of Edo and w. Indeed, one concludes that the demands on all aspects of the theory are negligibly reduced when one considers that the complete second harmonic response over a broad range of dc potential and frequency still must be accounted for by the theoretical formulation to interpret the data presented here. RESULTS AND DISCUSSION
The results of the second harmonic ac polarographic measurements of Cr(cN)~~-/cr(CN)6~in 1M KCN and CdZ+/Cd(Hg) in 1 M Na2S04are compiled in Figures 1 and 2, respectively. The solid curves represent theoretical polarograms corresponding to the values of k, and CY yielding the best overall agreement with experimental results. The relevant k , and CY values, as well as other parameters, are given in the figure legends. The points represent experimental second harmonic currents taken from automatically recorded second harmonic ac polarograms. The uncertainty in the ks and CY values yielding the best fit is about 50.02 cm sec-1 and i0.03, respectively-i.e., values outside this range yield unquestionably poorer agreement between theory and experiment. It is apparent that, with both systems studied, agreement between theory and experiment is generally excellent for the rate parameters selected. In both cases slight, but noticeable, disparities between theory and experiment occur at some frequencies. One is tempted to attribute some fmdamental significance to these differences, such as the onset of rate control by still a third process (e& a coupled chemical
Figure 2. Second harmonic ac polarographic results for Cd2+/Cd(Hg)system
System: 2.00 X 10-aM Cd2+in 1.00M N a ~ S 0 4 Applied: dc scan rate 10.0 mV per minute, 10.0 mV peak-to-peak sine wave: frequencieslisted below Measured : second harmonic faradaic component at end of mechanically controlled drop life. Readout aided by sample-and-hold circuitry. cm2sec-’, D R = 1.60 X 10-5 cm2 s a - ’ , -- Theoretical second harmonic polarograms for CY = 0.30, k, = 0.15 cm sec-’, D o = 6.00 X t = 5.40 sec., A = 0.0130 cm2,T = 25.0 “C, AE = 5.00 mV, n = 2, Co* = 2.00 X lo-”, and appropriate w value Observed second harmonic currents E. Applied frequency = 332.0 Hz A. Applied frequency = 23.0 Hz F. Applied frequency = 555.0 Hz B. Applied frequency = 46.0 Hz G. Applied frequency = 1110.0 Hz C. Applied frequency = 83.0 Hz H . Applied frequency = 2220.0 Hz D. Applied frequency = 166.0 Hz VOL. 41, NO. 1, JANUARY 1969
135
reaction), double-layer effects, etc. However, in view of the small magnitude of the deviations and the likelihood that they might simply reflect an occasional error in the calibration procedure, we feel that such speculation is without sound basis. The k , and CY values of 0.25 cm sec-’ and 0.55,respectively, for the Cr(CN)e*-/Cr(CN)a4- system are in excellent agreement with the values of 0.24 cm sec-’ and 0.59 recently obtained in these laboratories (12) by fundamental harmonic ac polarographic measurements, not to mention earlier reports of the k , value ranging from 0.22 cm sec-’ to 0.24 cm sec-1(10, 20, 22). Similarly, a recent fundamental harmonic ac polarographic investigation of the Cd*+/Cd(Hg) system (12) yielded k , = 0.15 cm sec-l and a = 0.30 which are identical to the values obtained in the present work. Comparable k , and CY values also were obtained in earlier studies (23-25). We view these results as extremely satisfying because not only is good agreement between second harmonic ac polarographic theory and experiment indicated for appropriate sets of rate parameters, but the apparent rate constants obtained are completely consistent with those obtained by other experimental approaches. Particularly significant in this regard is the establishment of self-consistency between rate parameters obtained from observables responsive to the linear aspects of the faradaic impedance (fundamental harmonic measurements) and those from observables controlled by
-
(22) J. E. B. Randles and K. W. Somerton, Trans. Faraday Soc.. ‘ 48,937 (1952). (23) J . K. Frischmann and A. Tirnnick, ANAL.CHEM.,39, 507 (1967). (24) H.~H.Bauer, D. L. Smith, and P. J. Elving, J. Amer. Chem. Soc., 82,2094 (1960). (25) G. Lauer and R. A. Osteryoung, ANAL.CHEM.,38, 1106 (1966).
faradaic nonlinearity. That the agreement is not fortuitous is suggested by the fact that, including the work of Randles and Whitehouse (6), three quasi-reversible systems have been found to date whose second harmonic response is satisfactorily interpreted by the appropriate second harmonic theory. NOMENCLATURE
A
= electrode area
D1
=
fi C,*
= activity coefficient of species i
Eo
= standard redox potential in European convention = dc component of applied potential
Edo AE
E1,g
=
=
diffusion coefficient of species i initial concentration of oxidized form
amplitude of applied alternating potential
= reversible polarographic half-wave potential (planar
diffusion theory) Faraday’s constant = ideal gas constant = absolute temperature n = number of electrons transferred in heterogeneous charge transfer step I(2wt) = second harmonic faradaic alternating current c $ ~ = phase angle of second harmonic current relative to applied alternating potential w = angular frequency t = time = heterogeneous charge transfer rate constant at Eo k, CY = charge transfer coefficient r = Euler Gamma Function m = mercury flow rate in mg sec-1 r, = electrode radius
F R T
=
Accepted October 2, 1968. Work supported by National Science Foundation Grant GP-7985. RECEIVED for review August 14, 1968.
Resistance Compensation in Nonaqueous Polarography. Evaluation of Uncompensated Resistance and Effects of Damping in Two- and Three-Electrode Cells William E. Thomas, Jr.,1 and Ward B. Schaap Department of Chemistry, Indiana University, Bloomington, Ind.
A simple, convenient method for calculating uncom-
pensated polarographic resistance is tested in aqueous and in glacial acetic acid solutions and in both twoelectrode and three-electrode cells. Equations are derived and verified which predict a difference between the half-wave potentials of heavily damped and undamped polarograms when uncompensated resistance is present. The sign and magnitude of the difference are dependent upon the relative magnitudes of the external and internal components of the overall cell resistance and are different for two- and threeelectrode cells. In the presence of ohmic distortion, linear extrapolation of the residual current is not satisfactory for the estimation of diffusion currents. A graphical method is presented which allows a better residual current line to be constructed by taking into account the distortion of the wave. RECENTADVANCES in controlled-potential instrumentation
have made possible more precise measurement of polarographic half-wave potentials in media of low dielectric constant and high specific resistance. Nevertheless, studies of 136
ANALYTICAL CHEMISTRY
high resistance polarography (1-4) have shown that, with media of high specific resistance or at high current densities, resistance compensation is not complete even with the use of three-electrode, potentiostat-type polarographs. Because the current flow in the immediate vicinity of the dropping mercury electrode appears to be spherically symmetrical, a significant potential gradient may exist in the direction of the reference electrode, even though the reference electrode is out of the path of the main electrolysis current. In the usual cell arrangement, the DME is placed between the reference and 1 Present address, E. I. du Pont de Nemours and Co., Orange, Texas.
(1) W. B. Schaap and P. S. McKinney, ANAL.CHEM., 36,29(1964). (2) Ibid.,p 1251. (3) W. B. Schaap and P. S. McKinney, “Polarography-1964,” Vol. 1, G . Hills, Ed., Macmillan and Co., London, 1966, p 197. (4) L. NemiZ, J. Elecrroanal. Chem., 8, 166 (1964).
