Some Experimental Studies on Time-Dependence of Alternating Current Polarographic Waves of Metal Ion-Metal Amalgam Systems Joseph
R. Delmastrol and Donald E. SmithZ
Department of Chemistry, Northwestern University, Evanston, Ill. 60201
The prediction based on stationary sphere theory that a.c. polarographic waves of metal ion-metal amalgam systems will exhibit a significant mercury column height dependence has been subjected to careful experimental test. The experimental results confirm the expectation that the column height dependence will be a general phenomenon with such systems. Various aspects of the observed column height dependence exhibited good qualitative agreement with predictions of stationary sphere theory, supporting the belief that the effect originates in spherical diffusion. However, the stationary sphere theory fares poorly in quantitatively reproducing experimental results, indicating the need for a more rigorous theoretical treatment com bined with carefully designed experimental procedures.
RECENT THEORETICAL WORK (I, 2) has suggested that a spherical diffusion-induced time-dependence of the a.c. polarographic current density should be a general phenomenon with metal ion-metal amalgam redox systems. The most conveniently observed and significant manifestation of this effect is a mercury column height dependence (drop-life dependence) of the a.c. polarographic currents ( I , 2). This phenomenon is predicted even for systems in which the a x . polarographic wave is diffusion-controlled, in contrast to the prediction of no column height dependence inherent in the classical a.c. polarographic theory (3-6). The observation of a mercury column height dependence of the a.c. wave has been reported for a few metal ion-metal amalgam systems (7, 8). In particular, Aylward and Hayes (8) observed a column height dependence with the Cd(II)/ Cd(Hg), Cu(II)/Cu(Hg), Pb(II)/Pb(Hg), and Tl(I)/Tl(Hg) systems which supports the prediction that this effect is a general one. The present paper describes results of a more extensive experimental investigation of the a x . polarographic column height dependence of metal ion-metal amalgam systems then previously reported. Included is a detailed comparison of the experimental observations with predictions of an approximate theory which is based on the stationary sphere electrode model. 1 Idaho Nuclear Corp., CPP, National Reactor Testing Station, P.O.Box 1845, Idaho Falls, Idaho 83401. To whom reprint requests should be addressed.
THEORY
The d.c. polarization process represents the origin of the spherical diffusion contribution to the a x . polarographic wave ( I , 2). The magnitude of the effect depends on the types of rate processes influencing the d.c. concentration profile (2, 9, IO). The simplest situation is the case where diffusion is the sole rate-determining step in the d.c. process. The present experimental study is concerned with the latter type of system, and only the theory relevant to this case will be examined. For the simple quasi-reversible electrode process, O+ne*R
in which charge transfer and diffusion are the rate-determining steps and the d.c. process is diffusion-controlled, the theoretical expression for the a.c. polarographic wave based on the stationary sphere electrode model assumes the form (2) Z(ot) =
I,,,F(t)G(w) sin (ut
+ 0)
(1)
where n zFzACo*(w Do) aAE Ire, =
F(t) = 1
4 R T cosh a( i / 2 )
+ DR"') + ( (DO1/' d D o 1 l a- DR1IZ)[ I
e = cot-'
(1
(2)
- exp (b*t) erf~(btl/~)](3)
+ T) &
(7)
D = DoOD,"
f = foafRR8 p = 1 - a (1) T. Biegler and H. A. Laitinen, ANAL.CHEM.,37, 572 (1965). (2) J. R. Delmastro and D. E. Smith, Zbid., 38,169 (1966). (3) B. Breyer and S . Hacobian, Australian J . Chem., 7,225 (1954). (4) M. Senda and I. Tachi, Bull. Chem. SOC.Japan, 28,632 (1955). ( 5 ) H. Matsuda, Z . Elektrochem., 61,489 (1957). ( 6 ) D. C. Grahame, J. Electrochem. SOC.,98,370 C (1952). (7) G. H. Aylward, J. W. Hayes, D. E. Smith, and H. L. Hung,
ANAL.CHEM.,36, 2218 (1964). (8) G. H. Aylward and J. W. Hayes, J. Electroanal. Chem., 8, 442 (1964). 1050
ANALYTICAL CHEMISTRY
(9) J. R.Delmastro and D. E. Smith, J. Electroanal. Chem., 9, 192 (1965). (10) D. E. Smith, ANAL.CHEM.,38, 347 (1966).
A = 8.515 x 1 0 - y m p 3
(13)
The notation employed in Equations 1-13 is identical to that used previously (2). A list of notation definitions is given in the Nomenclature. Equations 3 and 12 are applicable only to metal ion-metal amalgam systems in which the reduced form is initially absent in the electrode phase. The predicted influence of spherical diffusion and the accompanying mercury column height dependence are manifested only in the time-dependent F(t) term ( 2 ) . A combination of simple inspection and quantitative calculation makes apparent the following predictions of the foregoing equations for metal ion-metal amalgam systems characterized by diffusion-controlled d.c. processes ( 2 ) : (a) a spherical diffusion-induced mercury column height dependence is expected for any reasonable combination of Do and D E so that the effect is expected to be a general one; (b) the magnitude of the a.c. polarographic current always will increase with decreasing mercury column height (increasing drop life); (c) the magnitude of the column height dependence of the alternating current will exhibit a monotonic increase as the applied d.c. potential becomes more negative; (d) the magnitude of the mercury column height dependence will be independent of frequency; (e) the peak potential of the a x . polarographic wave will undergo a cathodic shift with decreasing mercury column height; (f) although the magnitude of the column height dependence is influenced by the magnitudes of Doand DR,the effect is small so that the general characteristics of the column height dependence will not vary significantly from one metal ion-metal amalgam system to another; (8) the magnitude of the column height dependence will be relatively insensitive to the n-value of the system; (h) with electrode processes which are diffusioncontrolled in both the d.c. and a x . sense (reversible waves), the magnitude of the a x . wave will be significantly influenced by spherical diffusion, but its shape will be relatively unaltered by this effect-e.g., reversible a.c. waves calculated from Equations 1-13 exhibit peak widths at half-height (halfwidths) of almost precisely 90jn mV and linear
us. E plots of i 118/n-mV slope which agree with predictions of planar diffusion theory (11); (i) the magnitude and peak potential of a x . polarographic waves of metal ionmetal amalgam systems will depend on capillary characteristics; (j) the phase angle will be independent of the effects of spherical diffusion-e.g., it will be independent of mercury column height.
EXPERIMENTAL
All a.c. polarographic measurements were carried out by phase-selective detection (11-14) of the alternating current component in phase with the applied potential. Theoretical predictions outlined in the previous section are unaltered by this mode of current detection. Potential control was effected with the aid of a solid-state operational amplifier potentiostat which was designed to eliminate all effects of ohmic potential drop when employed in combination with a three-electrode cell configuration. A detailed discussion of (11) D. E. Smith, in “Electroanalytical Chemistry,” A. J. Bard, Ed., Chap. 1 , Vol. 1, M. Dekker, New York, 1966. (12) J. W. Hayes and H. H. Bauer, J . Electroanul. Chern., 3, 336 (1962). (13) T. Takahashi and E. Niki, Tulantu, 1,245 (1958). (14) W. L. Underkofler and I. Shain, ANAL.CHEM., 37,218 (1965).
this device has been given elsewhere (15). The elimination of effects of ohmic potential drop combined with suppression of the double-layer charging current by phase-sensitive detection (11-14) resulted in a readout which could be taken as representative of only the faradaic current component. Direct current signal sources consisted of a precision ( ~ 0 . 1 % initial ) voltage source and a voltage ramp generator constructed from conventional operational amplifier circuits (16-18). A Hewlett-Packard Model 241A Sine Wave Oscillator was employed as the alternating voltage source. A IO-mV peak-to-peak sine wave was applied to the cell in all a.c. polarographic measurements. Most of the systems were studied at 20,40, and 80 Hz. Conditioning of the output signal of the potentiostat (the cell current signal) was effected by tuned amplification (19), phase-sensitive detection, and filtering the phase-sensitive detector output with a low-pass filter of second-order Butterworth response (20). An Electro Instruments Model 480 X-Y-Y’ recorder was utilized to record the output signal. A Hewlett-Packard Model 5243L Electronic Counter converted to a digital voltmeter with a Model 5263A plug-in provided for precise, high-resolution measurement of d.c. signals, which aided in proper adjustment of the phasesensitive detector. A Sargent Model S-29390 polarographic cell thermostated by an Aminco Model 4-8600 constant temperature bath was employed. Measurements were performed at 25.0 =t0.05” C. The polarographic cell and dropping mercury electrode (DME) assembly were electrically shielded by an aluminum case. In order to minimize problems due to mechanical vibrations, the cell and DME assembly were mounted independently on a steel frame which was bolted securely to the wall of the laboratory, A machined Teflon lid with holes drilled for insertion of electrodes and the nitrogen inlet/outlet served as the cover for the polarographic cell. The working electrode was a conventional blunt-tip DME made from Sargent S-29417 capillary attached to the mercury column with a nylon T-connector. A saturated calomel reference electrode and a platinum wire auxiliary electrode served to complete the cell. Oxygen dissolved in the polarographic solution was removed by bubbling with commercial Linde high-purity dry nitrogen gas. Before entering the polarographic solution, the nitrogen was passed through two gas bubblers filled with vanadium(I1) chloride solution followed by a bubbler containing the supporting electrolyte solution. In most of the studies reported here, one gas bubbler containing supporting electrolyte solution was sufficient to produce nitrogen with the same partial pressure of water vapor, etc., as in the polarographic cell. However, in the investigation of copper(I1) in l.OMNHa l.OMNHICI, three gas bubblers of supporting electrolyte solution in series had to be used to prevent significant loss of ammonia from the polarographic solution. During all experiments, a light stream of nitrogen gas was passed over the polarographic solution. Most of the compounds employed were reagent or primary standard grade. They were used without further purification, except for E. H. Sargent C.P. grade thallous nitrate which was twice recrystallized before use. Distilled water was passed through a charcoal filter bed (Continental Water of Chicago) before being used in solution preparation. Usually
+
(15) E. R. Brown, T. G. McCord, D. E. Smith, and D. D. DeFord, ANAL.CHEW, 38, 1119 (1966). (16) D. D. DeFord, Division of Analytical Chemistry, 133rd Meeting, ACS, San Francisco, April 1958. (17) M. T. Kelley, D. J. Fisher, and H. C. Jones, ANAL.CHEM., 31, 1475 (1959); 32, 1262 (1960). (18) W. M. Schwarz and I. Shain, Ibid.,35, 1770 (1963). (19) D. E. Smith, Ibid.,p. 1811. (20) Philbrick Researches, Inc., “The Lightning Empiricist,” Vol. 13, Nos. 1 and 2, Philbrick Researches, Inc., Dedham, Mass., 1965. VOL. 39, NO. 10, AUGUST 1967
1051
c
18.0
-
Ed c Epeak
Figure 2. Plot of d.c. potential VS. log
cFz)”*]
[(t)”’L \ ‘ I
from fundamental harmonic a.c. polaro-
grams of various metal ion-metal amalgam systems E d c ( V o l t s VS.
scE )
Figure 1. Fundamental harmonic ax. polarograms of Cu(I)/Cu(Hg) system at two different mercury column heights
+
System: 3 X lO-aM Cu(I1) in 1.OM NH3 1.OM NH4Cl Applied: 40.0-H~~ 10-mV peak-to-peak sine wave, d.c. scan rate 25 mV/min Measured : 40.0-Hz faradaic current component in phase with applied potential at end of natural drop life Drop life = 10.0 sec - _ _ _ Drop life = 5.26 sec the solutions were treated after preparation with special activated charcoal ( 2 1 ) and filtered immediately before use t o ensure removal of remaining traces of surface-active impurities. Systems chosen t o test the theoretical predictions were selected on the basis of literature reports ( 1 , 22-26) which characterized them as quasi-reversible systems with k , values which were sufficiently large so that the d.c. process could be considered definitely diffusion-controlled. With the low frequencies employed in these studies, most systems also could be considered diffusion-controlled in the a.c. sense. A list of the systems examined is given in Table I. The copper system involves a two-step reduction (22), the first yielding Cu(1) and the second Cu(Hg). The second step involving amalgam formation was of primary interest in this work. Calculations of theoretically predicted results for these systems utilized diffusion coefficients reported in the (21) K. M. Joshi, W. Mehl, and R. Parsons in “Transactions of the Symposium on Electrode Processes, Philadelphia, May 1959,” E. Yeager, Ed., Chap. 14, Wiley, New York, 1961. (22) D. S. Polcyn and I. Shain, ANAL.CHEM., 38, 370 (1966). (23) I. Shain and K. J. Martin, J . Phys. Chem., 65, 254 (1961). (24) W. G. Stevens and I. Shain, ANAL.CHEM., 38,865 (1966). (25) J. J. Lingane, ANAL.CHEM.,15,583 (1943). (26) J. E. B. Randles and K. W. Sornerton, Trans. Faraday SOC., 48, 951 (1952).
1052
ANALYTICAL CHEMISTRY
Systems:
Cd(I1) in 0.5M HCI Pb(I1) in 1.OM HClOd A = Cd(I1) inl.0 M KCl
0 =
0
=
+
+
0.1M NaC2H302 0.1M HCzH302 A = Pb(I1) in 1.OMKN03 Applied : 40.0-Hz, IO-mv peak-to-peak sine wave Measured : 40.0-Hz faradaic current component in phase with applied potential at end of natural drop life of approx. 10 sec Straight lines are of theoretical slope (59 mV) Table I. List of Systems Examined Experimentally References to Redox couple Supporting electrolyte previous studies CU(II)ICU(I), Cu(I)/Cu(Hg) Tl(I)/Tl(Hg) Cd(II)/Cd(Hg) Cd(II)/Cd(Hg) Pb(II)/Pb(Hg) Pb(II)/Pb(Hg) Bi(III)/Bi(Hg)
1.OMNH3 f 1 .OM NHdCl 1 .OM KNO3 f 0 . 1M NaC2H302 0 . 1M HCSH302 0 . 5 M HCl 1 .OM KCI 0.1M NaC2H302 0 . 1M HCzH302 1 , OM KNO, 1 . OM HClOd 1 . OM HC1
+
+
~
+
(22,251
(23) (1) (24
(26) (1) (26)
~~
literature (1, 23, 24, 27), except for Pb(I1) in 1.OM KNOI, Cu(I1) and Cu(1) in 1.OM NH3 1.OM NHIC1. Diffusion coefficients for the latter were calculated from d.c. polarographic limiting currents observed in first-drop experiments ( 2 8 ) with the aid of Koutecky’s equation (29).
+
(27) W. C. Cooper and N. H. Furman, J. Am. Chem. SOC.,7 2 , 5667 (1950); 74, 6183 (1952). (28) J. Kuta and I. Srnoler, “Progress in Polarography,” P. Zuman, Ed., with collaboration of I. M. Kolthoff, Vol. 1, Chap. 3, Interscience, New York, 1962. (29) J. Koutecky, Czech. J. Phys., 2, 50 (1953).
All data reported below refer to measurements of the instantaneous alternating current magnitude at the end of the natural mercury drop life. A large fraction of the data was obtained by current measurements at a fixed potential in which the current for a number of drops was recorded and averaged. This operation was repeated at various mercury column heights and at different, closely-spaced potentials along the a.c. wave, always using the maximum practical recorder sensitivity. This procedure allowed one to obtain more precise data than could be obtained directly from the recorded a.c. polarograms, particularly at the extremities of the a x . wave where currents are small. Errors due to slight d.c. drift in the amplifiers, reference electrode, etc., also were minimized through this approach. Measurements carried out in this work were confined t o the alternating current amplitude and, thus, the prediction that the phase angle would be independent of the mercury column height was not tested. However, data supporting this prediction have been obtained in other studies in progress in these laboratories (30). RESULTS AND DISCUSSION Essentially all qualitative predictions of the stationary sphere theory regarding the mercury column height dependerice of metal ion-metal amalgam systems were verified in this investigation. All amalgam systems exhibited alternating current amplitudes which increased significantly with decreasing column height (increasing drop life) at all d.c. potentials along the wave. A monotonic increase in the magnitude of this effect was observed as the d.c. potential became more negative. No noticeable effect of frequency was observed. Except for the thallium system, which appears to be an anomaly, the magnitude of the column height dependence and its variation with d.c. potential were nearly identical within experimental error for all systems studied, indicating that characteristics of individual systems such as Do, DR, and n normally play a relatively minor role in this effect. Despite the substantial column height dependence, allegedly manifesting a significant spherical diffusion contribution, the a.c. polarographic wave shapes were found to be in close agreement with planar diffusion theory for reversible systems-e.g., nearly 90/n-mV half-widths and linear
cs. E plots of 118/n-mV slope were observed. Finally, the peak potentials were found to exhibit a slight cathodic shift with decreasing mercury column height.
(30) E. R. Brown and D. E. Smith, unpublished work, Northwestern University, Evanston, Ill., 1966.
1.08-
LO4
Ed, (volts
VS.
SCE)
Figure 3. Ratio of fundamental harmonic current magnitudes at two different drop lives us. d.c. potential for Cu(I)/Cu(Hg) system
+
System: 3 x lO-3M Cu(I1) in 1.OMNHs 1.OMNHKl Applied: 0 = 20 Hz; A = 40 Hz; 0 = 80 Hz; 10-mv peak-topeak sine wave; d.c. potential varied incrementally Measured: Ratio of fundamental harmonic faradaic current components in phase with applied potential at end of natural drop lives of 10.0 sec ( t z ) and 5.26 sec (tl). Data depict average of currents measured over 5-10 drop lives at fixed d.c. potentials Other parameter values: m ( a ) = 0.764 mg sec-', m(t1) = 1.455 mg sec-l, Do = 9.69 X 10-8 cm2 sec-I, D R = 1.06 X cm2 sec-I, A (at end of drop life) = 0.0331 cm2 Solid curve represents theoretical values The basis for the foregoing statements is illustrated in Table I1 and in Figures 1-6. Table I1 gives observed and predicted cathodic peak shifts and half-widths, showing the reasonable agreement between theory and experiment for these parameters. No noticeable effect of frequency on this data was detected. Figure 1 shows a.c. polarograms of the Cu(I)/Cu(Hg) system at two different values of column height. These data on the copper system are in accord with the qualitative predictions of the stationary sphere theory and are typical of the results obtained with the other systems examined in this work. Figure 2 is a compilation of
c's. E plots obtained from the systems involving two-electron transfer. Figures 3-6 illustrate in a more quantitative manner than Figure 1 the general characteristics of the mercury column height dependence of the Cu(I)/Cu(Hg), T1(I)/T1(Hg), Cd(II)/Cd( Hg), and Pb(II)/Pb( Hg) systems with the aid of plots of the ratio of the alternating current at two different drop times us. d.c. potential. Such plots serve to highlight the spherical diffusion-induced time-dependence since, within the framework of the theory in question, devia-
Table 11. Cathodic Peak Shifts and Peak Widths a t Half-Height for Various Systems (Average of Data at 20,40, and 80 Hz) System Redox couple Cu(I)/Cu(Hg) TKI)/Tl(Hg) Cd(II)/Cd(Hg) Cd(II)/Cd(Hg) Pb(II)/Pb(Hg)
Supporting electrolyte 1 .OM "3, f 1.OM NH4Cl 1.OMKNO3 0.1MNaC2H302 0.1M HCgH302 0.5MHCl 1.OM KCl 0.1M NaC2H302 0.1M HCzH302 1.OM KNOa
+ + +
+
Cathodic peak shift, mV Observed Predicted 2.2 f 0.5 1.6
Peak width at half-height, mV, observed 92.0 f 0.5
2.0 f 0 . 5 0.8 f 0.3
0.85
90.5 f 0 . 5 46.0 =!= 0 . 4
0.9 =!= 0 . 2 1 . 1 f 0.4
0.80 0.70
45.6 f 1.0 48.0 i 1.0
1.7
VOL. 39, NO. 10, AUGUST 1967
1053
0
n 1.20-
s
o * b
.
F
4
I
h
8
a &
01.15-
1.20
. a
1.15.
A
6
;
8
A
.
9
8
a
.
o a
d 0 . 3
a
G II:
.
1.10-
1.05-
1.00
Figure 4. Ratio of fundamental harmonic current magnitudes at two different drop lives us. d.c. potential for Cd(II)/Cd(Hg) system System: 1 X lO-3M Cd(I1) in 1.OM KCI O.1M NaC2H302 0.1M HCzHaOz Applied: Same as Figure 3 Measured: Same as Figure 3 Other parameter values: Same as Figure 3 except D o = 7.35 X 10-6cm2sec-1, D R = 1.52 X 10-6cm2sec-l Solid curve represents theoretical values
+
+
tions of this ratio from unity can be attributed solely to this effect. The equations given above indicate that the theoretical expression for this ratio is: Ratio
Current at drop time tz Current a t drop time tl
Equation 14 was employed to calculate the theoretical values of this ratio depicted in Figures 3-6. These plots clearly illustrate the qualitative agreement of the experimental results with theoretical predictions regarding the effects of d.c. potential and frequency. We d o not believe the deviations of experimental points a t different frequencies shown in Figures 3-6 are representative of anything more significant than experimental uncertainty. The deviations in question are more or less random and they correspond t o relative average deviations from the mean of less than 1 of the total alternating cell current. Although the copper, cadmium, and lead systems (Figure 3-5) exhibit comparable behavior as theory predicts, the thallium system (Figure 6) shows a decidedly smaller column height dependence in better accord with quantitative theoretical predictions for the magnitude of the column height dependence. At this stage we believe the thallium system is best characterized as an anomaly and the fact that its behavior better reproduces quantitative theoretical predictions is probably fortuitous. This is based on the fact that all other systems studied produced results of the type shown in Figures 3-5 (substantial positive deviations from the theoretical curve) and on the already abundant evidence that the thallium process exhibits unusual behavior, probably due t o adsorption (31-35). Although the results depicted in Figures 3-6 are in accord with qualitative theoretical expectations, the quantitative agreement of theory and experiment is unsatisfactory. The observed mercury column height dependence normally is 1054
ANALYTICAL CHEMISTRY
1
Figure 5. Ratio of fundamental harmonic current magnitudes at two different drop lives cs. d.c. potential for Pb(II)/Pb(Hg) system System: 1 X I O - ~ M P ~ ( in I I 1.0MKN03 ) Applied : Same as Figure 3 Measured : Same as Figure 3 except tl = 5.56 sec Other parameter values: m(r2)= 0.703 mg sec-l, m(tl) = 1.266 mg sec-’, Do = 6.44 X 10-6cmz sec-’, D R = 1.16 X 10-6cm2 sec-l, A (at end of drop life) = 0.0312 cm2 Solid curve represents theoretical values
substantially larger than predicted by the stationary sphere theory. Thus, although this theory appears to yield correct qualitative predictions, it fails on a precise quantitative level. The fact that the observed column height dependence is larger than predicted should not be considered indicative of a gross contribution of spherical diffusion which exceeds theoretical expectations. It indicates only that the timedependence of the alternating current in the drop life range of 5-10 seconds exceeds the predictions of the stationary sphere theory. An independent observation indicates that the absolute magnitude of the observed spherical diffusion effect-i.e., the magnitude of the F ( t ) term-is smaller than predicted by the theory. This statement is based on observations with the copper system which exhibits two a x . polarographic waves, one involving a metal ion-metal ion redox couple [Cu(II)/Cu(I)] and one involving a metal ionmetal amalgam system [Cu(I)/Cu(Hg)]. Both a x . waves are essentially diffusion-controlled under the conditions employed in this work and d.c. polarographic measurements indicated that the diffusion coefficients of the Cu(1) and Cu(I1) ions are essentially equal. For such circumstances the stationary sphere theory predicts that the height of the a.c. wave due to the Cu(I)/Cu(Hg) couple will exceed the Cu(II)/ Cu(1) wave height by the factor, F ( t ) (Equation 3)--i.e., there will be no significant spherical diffusion effect on metal ion-metal ion a x . waves (2). Although the Cu(I)/Cu(Hg) wave height was substantially higher as predicted, its magnitude did not exceed that of the Cu(II)/Cu(I) wave by the amount predicted by Equation 3. The observed magnitude of this effect was about 6 and 14% for drop lives of 5.3 and 10 seconds respectively, while the predicted values were 17 (31) J. E. B. Randles in “Transactions of the Symposium on Electrode Processes, Philadelphia, May 1959,” E. Yeager, Ed., Wiley, New York, 1961. (32) R. Tamamushi and N. Tanaka, Z . Physik. Chem. N. F.,28, 158 (1961). (33) M. Sluyters-Rehbach, B. Timmer, and J. H. Sluyters, Rec. Truu. Chim.,82, 553 (1963). (34) P. Delahay and G. S . Susbielles, J . Phys. Chem., 70, 647 (1966). (35) G. S.Susbielles, P. Delahay, and E. Solon, Ibid.,p, 2601.
‘.*O
t
LO8C
I
-0.520
Edc
-0.480 VS.
(Volts
SC E )
-0.440
-0.400
Figure 6. Ratio of fundamental harmonic current magnitudes at two different drop lives us. d.c. potential for TI(I)/Tl(Hg) system
+
+
System: 3 X 10-3M TI(1) in 1.OM KNO3 0.1M NaC2H302 0.1M HCzH30z Applied : Same as Figure 3 Measured: Same as Figure 3 except t2 = 9.92 sec, tl = 5.31 sec Other parameter values: Same as Figure 5 except D o = 1.79 X 10-5 cm2 sec-l, D R = 9.9 X 10-8 cma sec-l, A (at end of drop life) = 0.0307 cm2 Solid curve represents theoretical values
and 24z respectively. Since the theory for reversible a x . polarographic waves of metal ion-metal ion systems appears to be reasonably precise ( ] I ) , the smaller than predicted difference in the a.c. wave heights of the copper system can be taken as indicating that the absolute magnitude of the spherical diffusion effect on the a x . wave of Cu(I)/Cu(Hg) is less than predicted by Equation 3. Additional evidence that the magnitude of the spherical contribution with metal ion-metal amalgam systems is less than predicted by the stationary sphere theory was found in second harmonic measurements (36). The results presented here together with previous reports (7, 8) definitely support the prediction that a significant spherical diffusion-induced mercury column height dependence generally will be operative in the a x . polarographic response of metal ion-metal amalgam systems, even with diffusioncontrolled processes. The qualitative characteristics of the column height dependence exhibit good agreement between stationary sphere theory and experiment, indicating that the phenomenon in question does originate in the spherical diffusion effect. However, the quantitative agreement between the stationary sphere theory and experimental results is poor. The theory apparently underestimates the magnitude of the column height dependence and overestimates the absolute magnitude of the spherical diffusion contribution. Such disparities between theory and experiment are not surprising as the stationary sphere calculation fails to consider other perturbing factors such as drop growth (29), shielding (36) T. G. McCord, E. R. Brown, and D. E. Smith, ANAL.CHEM., 38, 1615 (1966).
( 3 3 , depletion (28), and streaming (8, 38). It is apparent from the foregoing results that a precise theory for a.c. polarographic waves of metal ion-metal amalgam systems must incorporate the effects of spherical diffusion, but that the stationary sphere theory is quantitatively inadequate so that a more rigorous treatment is demanded. In addition, there is abundant evidence from d.c. polarographic studies (28) suggesting that even the most rigorous treatment of the expanding sphere boundary value problem presented to date (29) may not suffice to satisfactorily reproduce experimental results unless experimental conditions are designed to minimize shielding and depletion effects which are difficult to handle theoretically. To achieve this, the use of a pointed, tilted capillary and/or first drop experiments in place of the normally employed vertical blunt capillary and “serial” drops may be essential (28, 37). NOMENCLATURE
A
=
fi
= activity coefficient of species i
electrode area
diffusion coefficient of species i initial concentration of species 0 E” = standard redox potential in European convention AE = amplitude of applied alternating potential Ed.o. = d.c. component of applied potential El/{ = reversible polarographic half-wave potential (planar diffusion model) F = Faraday’s constant R = ideal gas constant T = absolute temperature n = number of electrons transferred in the heterogeneous charge transfer step Z(wr) = fundamental harmonic faradaic alternating current 6 = phase angle of fundamental harmonic faradaic alternating current relative to applied alternating potential w = angular frequency t = time k , = apparent heterogeneous rate constant for charge transfer at E” a = charge transfer coefficient r, = spherical electrode radius Zp = peak instantaneous fundamental harmonic faradaic alternating current at end of drop life Z = instantaneous fundamental harmonic faradaic alternating current at end of drop life m = mercury flow rate in mg sec-l Dt
C,*
= =
RECEIVED for review March 9, 1967. Accepted May 9, 1967. Taken in part from Ph.D. thesis of J.R.D., Northwestern University, 1967. Work supported by National Science Foundation Grant GP-5778. (37) H. Matsuda, Bull. Chem. SOC.Japan, 26, 342 (1953). (38) H. Strehlow and M. von Stackelberg, Z . Elektrochem., 54, 51 (1950).
VOL. 39, NO. 10, AUGUST 1967
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