Vibrating Dropping Mercury Electrode and Polarographic Theory Richard E. Cover and James G . Connery’ Department of Chemistry, St. John’s University, Jamaica, N . Y . 11432 The relationships between the response of the vibrating dropping mercury electrode (VDME) and fundamental electrode parameters such as drop time, rate of mercury flow, and frequency of vibration are examined. Close correlations with polarographic theory are found. Conformance to the llkovic equation by a mass-transport-controlled wave is observed over a wide range of experimental conditions down to millisecond drop times but the polarographic criterion for a diffusioncontrolled wave is inapplicable to the VDME. The current-potential curve for a reversible electrode process obeys existing theory over the entire range of conditions examined.
THE VIBRATING DROPPING MERCURY electrode (VDME) is essentially a DME in which the drop rate is controlled by periodic mechanical shock or vibration of the electrode. Such premature drop detachment has been used for various purposes over the last twenty years. In two previous papers ( I , 2), we demonstrated that, for most analytical purposes, the VDME is superior to the DME. When the vibrational frequency is sufficiently high, maxima of the first and second kinds can be eliminated at the VDME without the addition of surfactants. Furthermore, at high frequencies, catalytic and kinetic waves can be minimized or eliminated from VDME response. Waves of these types are rarely useful analytically and can obscure desired data. The VDME also permits the analysis of agitated solutions. In addition, the analytical behavior of the VDME is superior to that of the DME when the adsorption of substances on the electrode surface can inhibit response. Such inhibition can affect electrode response adversely in several ways. Detector response may be decreased. The polarographic waves may be grossly distorted preventing meaningful current measurements or the electrode response to concentrations may become nonlinear. At the VDME under the proper conditions, these adsorption effects can often be minimized or eliminated and analytical data obtained. This improved analytical behavior makes theoretical understanding of the VDME worthwhile. In the work reported here, the relationships between VDME response and fundamental parameters such as drop time, rate of mercury flow, and the frequency of electrode vibration are examined. Because the hydrodynamics of the VDME are unquestionably very complex, no attempt was made to derive a theory for this electrode based on a detailed consideration of the transport phenomena involved. Instead, correlations have been sought between VDME behavior and existing theory of polarography and the electrolysis of stirred solutions. The ranges over which the effects of electrode parameters have been studied are: drop time, 2.7 X to 3.1 seconds; frequency, 0 to 210 Hz; and rate of mercury flow, 2.1 to 7.6 mg sec-I.
EXPERIMENTAL Apparatus. All drop time measurements at the VDME were made oscillographically using an Electronics Associates Model 34.035 Repetitive Operation Display Unit and a Model 800 Polaroid camera. The time-sweep source was a comparator-latch circuit programmed on an Electronics Associates TR-20 analog computer. Each photograph was calibrated with a superimposed image of known time duration. All drop time and rate of mercury flow measurements were made at -0.8OV US. SCE. All initial and final potentials on the polarograms and voltammograms were measured to iO.5mV with a Leeds and Northrup Model 7655 potentiometer. All other apparatus used were previously described ( 1 ) . Reagents. All reagents used were Baker or Mallinckrodt reagent grade. RESULTS AND DISCUSSION
Present address, Leeds & Northrup Co., Technical Center, North Wales, Pa. 19454.
The reduction of Cd(I1) in 0.1M K N 0 3is well known to occur reversibly and to be diffusion-controlled under polarographic conditions, For these reasons, this system was chosen to evaluate VDME response. All polarograms were run at a cadmium concentration of 3.97mM; at this level, no contribution from migration current was detectable. Voltammograms were run and drop time and rate of mercury flow measurements were made at eight different frequencies and four different mercury pressures for a total of 32 separate experiments. Under certain conditions, drop times were not reproducible even though good voltammetric responses were obtained. These poor data precluded the calculation of parameters such as diffusion-current constants and rates of electrode area formation. Eight data points were so rejected. At the two highest mercury pressures at frequencies below 154 Hz, maxima of the second kind were frequently observed. This phenomenon prevented the accurate calculation of such parameters as limiting currents, half-wave potentials, and diffusion-current constants. Nine data points were rejected on this basis. Limiting Currents. The Ilkovic equation predicts that the limiting current for a diffusion-controlled process should be proportional to the product ( m z / 3 f 1 / 6 )where m is the rate of mercury flow in milligrams sec-I, and r is the drop time in seconds. Figure 1 contains a plot of data obtained for frequencies from 0 to 210 Hz and drop times as low as 2.7 msec. The limiting currents plotted are mean currents measured from voltammograms by conventional extrapolation techniques. As can be seen, all the DME and VDME data comply reasonably well with a linear relationship of the same form as the Ilkovic equation. The observed slope, however, is considerably larger than that predicted for mean currents by theory (3). Furthermore, multiple regression analysis of the VDME data indicates conformance with this relationship. Graphical inspection indicates that the DME and VDME data can be legitimately considered as members of the same
(I) R. E. Cover and J. G. Connery, ANAL.CHEM., 41,918 (1969). (2) J. G. Connery and R. E. Cover, ibid.,p 1191.
(3) L. Meites, “Polarographic Techniques,” 2nd ed., Interscience Publishers, New York, 1965, pp. 111-25.
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
1797
40
where h is the total mercury pressure and and P are constants. These constants have been successfully related to capillary dimensions and the properties of mercury and the electrolyte solutions (6) assuming spherical electrode geometry. Our data for both DME and VDME behavior obtained with a particular capillary for 3.97mM Cd(I1) in 0 . 1 M K N 0 8 were found to fit this equation at all frequency levels.
I
m = (0.0493 i 0.0007 mg sec-’ cm-l) (h
- (3)
(2)
The back-pressure correction, (3, is theoretically (7) the sum of two terms: a small hydrostatic contribution due to the immersion of the capillary tip in the solution and a larger term due to the interfacial tension between the mercury and the solution. In our case, then, theoretically
P I
I 2
I m2’l
I 3
,1/6
Figure 1. Limiting currents and the Ilkovic equation Solid points, DME data obtained with three different capillaries, Open circles, VDME data (a) Least-squaresfit for all data (b) Theoretical curve, taking D = 7.6 X cma seC-1
statistical population although the average diffusion-current constants of 3.99 + 0.04 for the DME and 4.19 & 0.28 for the VDME leave the question open to some debate. The fact that DME and VDME response are related in any fashion is surprising because the convection surely obtaining at the VDME violates assumptions fundamental to the Ilkovic equation. The high experimental slope obtained reflects the fact that the diffusion current constants for both DME and VDME are high relative to the theoretical value of 3.37. The direction and magnitude of these discrepancies are consistent with the data of Heyrovsky and Kuta (4). Our data do not show the sharp increase in diffusioncurrent constant found by other workers (3) at the DME at drop times below two seconds. This may be due in part to the fact that we collected no limiting current data at drop times between 0.01 to 1.0 second. On the other hand, Wolf‘s data (5) obtained with a VDME at drop times in this range, did not show this effect either. The increase in diffusion-current constant at short drop times at the DME is attributed to motion of the solution near the capillary tip. Perhaps the convection due to electrode vibration at the VDME is sufficient to render negligible the stirring effect due to rapid drop fall. Previous work ( I ) on the uranium-catalyzed reduction of nitrate ion and the kinetic wave of formaldehyde indicates that existing theory valid for these processes at the DME is applicable to a good approximation to the VDME as well. Rate of Mercury Flow. The rate of mercury flow, m,from a capillary under polarographic conditions has been demonstrated to obey an equation of the form m = cu(h - (3)
e
+ 3.1 (mr)-l’a
(3)
Table I summarizes the capillary back-pressure correction data. At the polarographic level (0 Hz), theory is obeyed to within experimental error. As the frequency of vibration is increased, the back pressure corrections become larger as does the deviation from theory. The cause for this deviation apparently lies in the assumption of spherical geometry made in the derivation of the second term of Equation 3. Implicit in the derivation is the assumption that the drop is spherical throughout drop life (6). While clearly this cannot be true during the early stages of drop formation, at the DME, the drop is indeed spherical throughout most of its life because final drop diameters are about 15 times larger than the capillary orifice diameter. At the VDME, however, drops were observed which were only twice as large as the capillary orifice. Conformance to the geometrical assumptions cannot be expected for such a situation. The data in Table I support this conclusion. As displacement frequency is increased, smaller drops are obtained and deviations from theory become larger. The direction of the deviations, that is, theoretical back pressures are less than observed ones, indicates that the mean VDME surface areas are less than those predicted from sphericity assumptions. Limiting Currents and Mercury Pressure. A simple test which has long been used as a criterion of a diffusion-controlled process in polarography is based on the equation id = k hoorrl‘z
(4)
where hoorr= h
-P
(5)
Equation 4 can be readily derived from the Ilkovic equation on the basis of Equation 1 and the assumption that drop times are inversely proportional to h,,,, (4). Conformance of limiting currents of a polarographic wave to Equation 4 indicates strongly that the electrode process is diffusion-controlled. As the data in Table I1 indicate, this criterion for mass transfer control cannot be extended to the VDME. This observation, however, is not surprising. Even though the experimental data do conform to both Equation 1 and the Ilkovic equation, the observed drop times are not inversely proportional to h,,,, but are a function of displacement frequency as well.
(1)
(4) J. Heyrovsky and J. Kuta, “Principles of Polarography,” Academic Press, New York, 1966, pp 85-6. (5) D. Wolf, J . Electroanal. Chem., 5, 186 (1963). 1798
= 0.07 cm
(6) I. M. Kolthoff and J. J. Lingane, “Polarography,” 2nd ed., Interscience Publishers, New York, 1952, pp 78-86. (7) L. Meites, “Polarographic Techniques,” 2nd ed., Interscience Publishers, New York, 1965, pp 131-2.
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
Table I. Capillary Back Pressure Corrections, cm Hg Frequency, Hz Experimental Theoretical 0 42 70 98 126 154 182 210
O
I
20
I
I 60
A0
80
i / D r o p D i a m e t e r , cmr’
Figure 2. Dependence of cell resistance on drop diameter Frequency range 0 to 210 Hz Charging Current. As Meites has pointed out (8), for constant rates of mercury flow, the charging current should be proportional to t-lI3. Thus, for a thousand-fold decrease in drop time, the charging current should increase by a factor of ten. Comparing VDME and DME data where m did not vary widely, the charging currents did increase over the DME levels by an order of magnitude when the drop time was in the millisecond range. While no precise test of theory was made, this point must be explicitly clarified because the larger charging currents at the VDME affect its sensitivity as an analytical detector. Current-Potential Curves. Theoretical equations have been derived (9) for current-potential curves obtained with stirred solutions or at moving electrodes on the basis of the Nernst diffusion layer concept. According to Delahay, both reversible and irreversible processes should obey the equation
E = a - b l o g ( & JI 1 -
I
where a and b are constants which are related to fundamental parameters, i is the current at potential E and il is the limiting current. The conformance of VDME behavior to this equation was examined. Because the voltammograms obtained with our equipment are actually current-voltage curves, corrections for iR drop are essential. The resistance entering into these corrections is the sum of the external circuit resistance, which can be easily computed, and the cell resistance. The evaluation of the cell resistance at the VDME presents a real problem. Under polarographic conditions, where DME behavior is well understood and electrode geometry is well defined, dependable means for determining cell resistance are available (IO, 11). In the case of the VDME, however, viable theory does not exist, drop diameter varied by an order of magnitude over the range of our experiments and the electrode geometry is highly questionable. Cell resistance is, thus, likely to vary with experimental conditions at the VDME. Furthermore, (8) L. Meites, “Polarographic Techniques,” 2nd ed., Interscience Publishers, New York, 1965, p 100. (9) P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience Publishers, New York, 1954, pp 217-26. (10) L. Meites, “Polarographic Techniques,” 2nd ed., Interscience Publishers, N. Y.,1965, p 70. (11) J. Heyrovsky and J. Kuta, “Principles of Polarography,” Academic Press, New York, 1966, pp 61-4.
1.80 4.67 7.53 8.69 10.87 14.77 16.52 17.30
f 0.57 Z!Z 0.21 i.0.19 f 0.61 =k 0.63 f 0.97 =k 0.39 i 0.66
1.58 f 0.00 4.09 5 . 3 8 & 0.16 6.13 f 0.42 8.14 i. 0.19 11.34 i 0.88 12.43 & 0.88 12.57 i 1.37
Table 11. Limiting Currents and Mercury Pressure, Values of p in the Equation il = k h P C o I r Frequency, Hz P 0.89 + 0.07 0.62 It 0.005 0.65 i.0.004 0.67 f 0.11 0.57 i 0.09 0.59 i 0.12 0.72 f 0.08
42
IO 98 126 154 182 210
this resistance cannot be simply computed nor directly measured. The technique used here is based on these assumptions: (a) that Equation 6 is a valid model for a VDME voltammogram and (b) that the actual total cell-circuit resistance is that value of R which gives the best fit of the data to Equation 7, V + iR
=
a
- blog ( i, i)
(7)
where V is the applied voltage. The calculations were performed on a digital computer where the criterion of goodness of fit was taken as the unweighted sum of squares of deviations between the observed and calculated dependent variables. For each voltammogram, a set of seven data points was obtained in the interval 0.1 il < i < 0.9 il on the rising portion of the wave. R was taken initially as the external circuit resistance, R1, and an unweighted least-squares fit made to the data. Rt was taken as (R1 200 ohms) and a second fit was made. If the sum-of-squares criterion indicated an improved fit, R was further incremented. This process was continued with Rm+l = (R, 200 ohms) until the goodness of fit decreased whereupon the increment was changed to -0.5 of the previous increment. This computation was continued until J R , Rn+lI 5 0.05 ohm. The refined resistance estimate was printed and the calculations were terminated. The consistency between DME and VDME results are striking: a = Eliz = -0.586 f 0.002 V for the DME and a = Eliz = -0.587 i.0.003 V for the VDME. The values of b are 29.7 + 0.6 mV for the DME and 29.1 f 1.9 mV for the VDME. The results are based on DME data taken from three different polarograms while the VDME data were obtained from 17 different voltammograms in the frequency range from 70 to 210 Hz with drop life ranging from 2.70 to 60.8 msec. The value of Elp given by Meites (12) for Cd(I1) in 0.1M KNOa is -0.58 V. The theoretical value forb for a reversible two-electron reduction is 29.6 mV at 25 “C.
+
+
(12) L. Meites, “Polarographic Techniques,” 1st ed., Interscience Publishers, New York, 1955, p 255.
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
1799
!+
60
P P -1
‘n 0-
;v
- 0.05
0001
I 50
I
I
100
I50
I 1 200
D i r p l o c a m e n t Frequency. H i .
Figure 3. Dependence of rate of area formation on frequency of vibration Total mercury pressure, 159.6 cm
These data indicate very strongly that Delahay’s equation for a reversible process is valid for this system at the VDME and that Cd(I1) is reduced reversibly even at millisecond drop times. Furthermore, these data strongly support the method by which the cell resistances were evaluated. Further support for this resistance computation can be obtained by correlation of the estimated cell resistance with drop dimensions. Ilkovic (11) showed that cell resistance is inversely proportional to the electrode diameter for a spherical electrode in a conical conductional path. These geometrical constraints are approximately satisfied between the VDME and the end of the H-cell salt bridge. Figure 2 contains a plot of cell resistance determined by the data fitting technique os. the reciprocal of the drop diameter. (Drop diameters were calculated from m and t measurements assuming spherical electrode geometry.) The data do fit a linear relationship reasonably well. Rate of Area Formation. One of the most important characteristics of the VDME is the high rate of electrode area formation relative to the DME. This property permits superior analytical response in the presence of adsorption phenomena (2) because it operates to minimize the extent of surface coverage during detector life. Figure 3 shows the relationship between rate of area formation and frequency of vibration. The rate of area formation was estimated by calculating maximum area per drop (assuming drop sphericity) and dividing by the drop time. It is, therefore, an average estimate. As the data indicate, the rate of area formation increases by about an order of magnitude in going from 0 to 210 Hz. This phenomenon is, of course, responsible for the high charging currents observed at the VDME. Drop Time and Frequency. Drop time is a continuous function of frequency of vibration as can be seen from the data in Figure 4, although the notion which seems prevalent in the older literature, that a simple relationship exists between these parameters, is clearly not correct. As curve (b) in Figure 4 indicates, mechanical shock is not the only parameter which affects drop detachment. While it is true that some data points indicate drop detachment at the extremes of electrode displacement, detachment occurs at other points in the traverse as well. As can be seen from Figures 3 and 4, in the region from 126 to 154 Hz, sharp increases are observed in both the rate of area
1800
,.
I
I
I
50
100
150
D i s p l a c e m e n t Frequency,
1
ZbO
lo
Hz
Figure 4. Dependence of drop life on the frequency of vibration Total mercury pressure, 159.6 cm (a) Drop time us. frequency (b) Drops per cycle us. frequency
formation and the number of drops per cycle. This behavior is related to the hydrodynamics of the system. When the Reynolds number (Re) is calculated from our data based on the movement of a sphere through water (I.?), it is found that, in this region, Re increases with frequency to about 10. At this level of Re, significant deviations from Stokes’ law occur and laminar flow definitely obtains. Furthermore, in this region the drag force on the spherical electrode increases by about SO%, thus enhancing the tendency to shear mercury droplets from the capillary. As indicated by curve (b) in Figure 4, at 210 Hz, the number of drops per cycle decreases from the maximum observed at 182 Hz. This behavior is probably due to the inescapable fact that, for a given mercury pressure and capillary orifice, there must be a minimum possible drop size. At both 182 and 210 Hz, the drop diameters were 140 p and correspond to a drop to capillary orifice diameter ratio of 2.1. In principle, regular drop formation could be obtained down to a ratio of 1.0 but this does not seem to occur. In their studies with the rotating dropping mercury electrode, Okinaka and Kolthoff (14) observed that at Re = 230, the ratio of drop to orifice diameters in nine experiments with three different capillaries was 2.0 i 0.1. At this time, there is no sound theoretical basis for this ratio but the comparison of results is striking. Thus, above 182 Hz level, it appears that the drop time is primarily determined by the mercury flow rate and a further increase in frequency will only result in a decrease in the number of drops per cycle. RECEIVED for review June 6, 1969. Accepted August 27, 1969. Presented in part before the Division of Analytical Chemistry, 158th National Meeting, ACS, New York, N. Y., September 8, 1969. The following support made this work possible: a faculty summer research stipend from St. John’s University for R. E. Cover and a 1968 Summer Fellowship Award from the ACS Division of Analytical Chemistry for J. G. Connery. (13) R. B. Bird, W. E. Stewart, and E. N. Lightfoot, “Transport Phenomena,” J. Wiley and Sons, New York, 1960, pp 182, 190-4. (14) Y. Okinaka and I. M. Kolthoff, J. Amer. Chem. SOC.,79, 3326 (1957).
ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969
