(9) Kinoshita, &I.,Bull. Chem. SOC.Japan 35, 1609 (1962). (10) Klemm. L. H.. Reed, D.. Miller, L. A., Ho, B. T., J . Org. Chem. 24, 1468 (1959). (11) LeRosen, A. L., Monaghan, P. H., Rivet, C. A,, Smith, E. D., ANAL. CHEM.23, 730 (1951). (12) Smets, G., Balogh, V., Castrille, Y., J . Polymer S a . , pt. C, 4, 1467 (1964).
(13) Spotswood, T., Australian J. Chem. 15, 278 (1962). (14) Streitwieser, A., J . Am. Chem. SOC. 82, 4123 (1960). (15) Tye, R., Bell, Z., ANAL. CHEM.36, 1612 (1964). (16) Watanabe, K., J . Chem. Phys. 2 6 , 542 (1957). (17) Wheland, T. W;,"Resonance in Organic Chemistry, pp. 75-152, John
Wiley and Sons Inc., New York, N. Y., 1955. RECEIVED for review October 7, 1965. Accepted March 21, 1966. Work supported under Grant A 536 from the Petroleum Research Fund and Grant GM 10064 from the U. S. Public Health Service, National Institutes of Health.
EIectroIysis with Constant Potential Diffusion Currents for the Formation of an Amalgam at a Hanging Mercury Drop Electrode WILLIAM G. STEVENS and IRVING SHAIN Chemistry Department, University of Wisconsin, Madison, Wis.
b The effect of the spherical nature of the hanging mercury drop electrode on the diffusion current of a reaction in which an amalgam is formed has been verified for the potentiostatic reduction of Cd(ll) near the formal E" of the system Cd(ll)/Cd(Hg). The spherical correction term to the diffusion current equation changes sign, and the experimental results are in excellent agreement with the theory.
I
previous study of the potentiostatic reduction of thallous ion at a hanging mercury drop electrode ( 7 ) , it was observed that a t potentials near the formal E" for this reversible system, the magnitude of the spherical correction term in the characteristic currenttime equation varied; and a t the formal E", the spherical correction term changed sign. Thus, a t potentials cathodic of E " , the observed current was higher than would have been expected for a plane electrode of the same area, while a t potentials anodic of E", the currents were smaller. It was suggested that this was related to the convergent nature of the diffusion process within the spherical mercury electrode, which would lead to higher concentrations of the thallium amalgam a t the surface of the electrode than would have been obtained for a linear diffusion case. This suggestion was qualitatively in accord with an approximate limiting solution to the boundary value problem which was obtained by considering the convergent diffusionprocess within the electrode under conditions where the amalgam concentration is very low. A more general solution could not be obtained because of the complexity of the mathematics encountered when an attempt was made N A
to account for both the convergent diffusion and the finite volume of the drop. Recently, in a careful study of the effect of the formation of an amalgam in ax. polarography, similar spherical correction term effects were observed by Delmastro and Smith ( 3 ) . They published a rigorous solution to the boundary value problem for a.c. polarography in which the convergent diffusion was considered but the finite volume of the drop was ignored, an approach first suggested by Reinmuth (6). The equations describing the a.c. polarography experiment include the result for a constant potential experiment, and an explicit equation for the potentiostatic case can be obtained by taking Equations 1 and 50 from Reference (S), and combining them with the Nernst Equation for the reaction 0 ne2R:
+
nFADoCO*[l/(l l/ro(l - 7%) - e(?
i
=
+ re)+ + 1)2(e@'terfc x
d & - - m ( Y~ e + 1121
(1)
a result obtained independently in this laboratory (10). I n Equation 1,
P
- i ) / r(7e ~ ~+ 1) = @Do - D R ) / T o ( e d D i + fi) =
-\/oo(-t2e
and all other terms are as defined previously (7, 8). It should be noted t h a t Equation 1 is the same as that derived previously (8) for the case in which both 0 and R are soluble in the solution, except for the different signs in the second and third terms in the brackets. There are several limiting forms of Equation 1 which are of importance. First, for very cathodic potentials, e+, and Equation 1 reduces to
i = nFADoCo*[l/d*Dot
+ l/r,,]
(2)
which is the result expected for this case where diffusion of substance 0 is the controlling factor in the rate of the reaction ( 2 ) . On the other hand, for a reduction taking place a t potentials significantly anodic of E " , 0 becomes large, and since the third term in brackets in Equation 1 is a function of 1/02, while the second is a function of l / O , the third term becomes small more rapidly than the second. Under these conditions, Equation 1 reduces to
i
nFdDoCa* jl/yOdrDOt -
=
l/-P0roI
(3)
Equation 3 is of the same form as the result obtained previously on the basis of qualitative considerations [Equation 14, in Reference ( 7 ) ] . The third case of interest is a t potentials near the E" for the system, since it is in this region that the spherical correction term changes sign as predicted previously from experimental results ( 7 ) . From Equation 1, the spherical contribution to the current is time dependent, and in general, for any particular potential in this range, the spherical correction term decreases as a function of time. This can be evaluated from Equation 1 by expanding the t p d t as a series, function exp ~ * erfc using the form valid for small values of the argument (since the argument never exceeds unity for realistic values of the experimental parameters). When the first two terms of the series are used, the spherical correction term, u-i.e., the last two terms in Equation 1-becomes:
+ ro)zl[l - 0 -O(y + x 2d%/7rodi(1 + (4)
u = [l/To(l 0
2
TO)]
At t = 0, u is positive a t potentiah cathodic of E O , and negative a t potenVOL 38, NO. 7, JUNE 1966
8
865
Vollr
Figure 1 . Time and potential dependence of the vanishing point of the spherical correction term
tials anodic of E". In all cases, u decreases with time; the time dependence being the largest at E'. The time dependence of the point a t which the spherical contribution to the current vanishes can be obtained from Equation 4 by setting u = 0: d%/r,
=
d&l- e)
(1
x
+ re)/2e(1 +
712
(5)
This relation between the potential, the ratio of the diffusion coefficients, and the parameter d & t / r o is shown in Figure 1. Any specific set of experimental parameters will determine a point on this graph. If the point lies above the line for the appropriate value of 7,u will be positive, if below, u will be negative. I n the normal case, the potential dependence of potentiostatic experiments is demonstrated by plotting the current flowing a t some fixed time after the start of the electrolysis as a function of potential. The point a t which the spherical correction term vanishes will always be cathodic of E o in such a plot, and the exact potential can be obtained from Figure 1. To verify these theoretical relations, extensive potentiostatic studies were carried out on the reduction of Cd(I1) in 1.0kf potassium chloride, 0.1M sodium acetate, 0.1M acetic acid, pH 4.7. EXPERIMENTAL
All experiments were carried out on an instrument based on operational amplifiers. The circuit was the same as that described previously in Figure 9 of Reference ( I S ) , except that the controller amplifier did not include a booster amplifier. I n addition (using the previous notation) the analog integrator, I , and the associated recorder, M-2, were not used. PS-1 and PS-2 were battery-operated low voltage power supplies. M-1 was a Leeds and Northrup Speedomax G Series 6000 recorder with a chart speed of 27.9 0.2 inches/minute, a full scale sensi-
*
$66
ANALYTICAL CHEMISTRY
VI.
Ag/AgCI
Figure 2. Current flowing 5 seconds after start of electrolysis. Comparison of plane and spherical electrodes Solution: 1.00 X 10-3M Cd(ll) in 1.OM KCI, 0.1M N a acetate 0.1 M ocetic acid Electrode radius, 0.0659 cm. Points, experimental; solid line, theoreticol for a spherical electrode (Equation 1); dashed line, theoretical for o plane electrode
tivity of 10 mv., and a nominal full scale response of 0.4 second. The cell was similar t o that described previously ( I ) , except that the holes in the Teflon lid were provided with neoprene "0" rings to help keep out air. The counter electrode was 2 inches of 18-gauge platinum wire, mounted on the mercury transfer scoop. The reference electrode was Ag-AgC1, placed directly in the cell. All experiments were carried out a t 25.0' f 0.1" C., and the cell and thermostat were mounted on a 60-lb. sIab of concrete (supported by semi-inflated plastic balls) to minimize vibration. High purity nitrogen was passed through a vanadous sulfate solution, washed, and then passed through each solution for a t least 15 minutes to remove oxygen. All chemicals were reagent grade and were used without further purification. The electrode area was determined after each experiment by collecting the individual hanging mercury drop and weighing it on a micro balance. The working electrodes were made from 6-mm. glass tubing and were provided with convection shields as described previously ( I ) . To minimize the effects of shielding of the electrode by the glass (7), the tip of the electrode was carefully ground to a point, so that very little glass remained around the platinum-glass seal. In all calculations, the area of the electrode was corrected by subtracting that portion subtended by the platinum and the glass support (generally of the order of 0.0013 sq. cm.). RESULTS AND DISCUSSION
Current-time curves were obtained for the reduction of 10-3i!f Cd(I1) solutions over the potential range from -0.5 v. to -0.7 v. us. Ag-AgC1, using hanging drop electrodes ranging from 0.052- to 0.076-cm. radius. By using the equipment as described, useful data were obtained over the time interval
from 1 to 30 seconds. The data were treated by plotting ~ ~ / ~ / T L F as A Ca * ~ function of dt, as suggested by Lingane (4). Straight lines were obtained in each case, with positive slopes at potentials more cathodic than about -0.61 volt us. Ag-AgC1, and negative slopes a t more anodic potentials, as predicted by Equation 1. From curves obtained at very cathodic potentials (-0.725 volt) the diffusion coefficient of Cd(I1) in IJP potassium chloride, 0.1M sodium acetate, 0.1M acetic acid was determined to be 0.735 X 10-5 sq. cm./sec., with a relative standard deviation of 2.8%, calculated from both the slope and the dtus. dtcurves. This intercept of the i value is somewhat lower than that reported by Skobets and Kavetskii (9), although other values have been reported ranging from 0.700 to 0.781 X ( 5 , 1 2 ) . The value reported here is considered reliable within the indicated level of accuracy, since the effects of convection and other uncontrolled variables were minimized. The correlation between values of D calculated from slope and intercept agreed closely, after the area of the electrode was corrected for the portion shielded by the glass and platinum of the support. When this value of the diffusion coefficient was used, the experimental results were correlated with Equation 1. The excellent agreement is shown in Figure 2 where the current, flowing five seconds after the start of the electrolysis, is plotted as a function of potential. For comparison, the current expected for a plane electrode is also shown, demonstrating the sign change for the spherical correction term a t a potential slightly cathodic of E'. It was previously pointed out (8) that for the case in which the product of
the electrode reaction is soluble in the solution, it is possible to obtain independent values of y0 and y20 from current time curves taken near the formal E o for the system. Since DO can be determined separately from data obtained at very cathodic potentials, both 0 and DR can be obtained from cathodic experiments only. When substance R is soluble in the mercury, it is possible to make the same calculations using Equation 1. These calculations were carried out for the Cd(I1)-Cd(Hg) system used in this work, and a value of 2.303(RT/nF) log 0 = 0.606 volt US. Ag-AgC1 was obtaiwd. For DB, a value of 1.6 X sq. cm./sec. was obsq. tained, compared with 1.61 X cm./sec. obtained from direct anodic
dissolution of cadmium amalgams (11). This procedure should make it possible to calculate 2.303(RT/nF) log 0 to =t2 mv., and D R to about 5% from reduction experiments only. The method should be useful in those cases where it is not feasible to prepare stable amalgam electrodes for direct study. LITERATURE CITED
(1) Alberts, G. S., Shain, I., ANAL. CHEM.35,1859 (1963). (2) Delahav, P., “New Instrumental Methods” in Electrochemistry,” p. 61, Interscience, New York, 1954. (3) Delmastro, J. R., Smith, D. E., ANAL.CHEM.38, 169 (1966). (4) Lingane, P. J., Zbid., 36, 1732 (1964). (5) Macero, D. J., Rulfs, C. L., J . Am. Chem. SOC.81, 2942 (1959).
(6) Reinmuth, W. H., ANAL. CHEM.33, 185 (1961). (7) Shain, I Martin, K. J., J . Phys. Chem. 65.254 11961’). (8) Shain, ’I., Polcyn, D. S.,Zbid., 6 5 , -1849 ---
(1981). \ - - - -
(9) Skobets, ’E. M.,Kavetskii, Tu’. S., Zh. Fiz. Khim. 24, 1486 (1950). (10) Stevens, W. G., Ph.D. Thesis, Uni‘ versity of Wisconsin, 1966. (11) Stevens, W. G., Shain, I., J . Phys. Chem. (in press). (12) Turnham, D. S., J . Electroanul. Chem. 10, 19 (1965). (13) Underkofler, W. L., Shain, I., ANAL. CHEM.35,1778 (1963).
This work was performed in part while W. G. Stevens held an NSF predoctoral fellowship, and also was supported in part by funds received from the U. S. Atomic Energy Commission under Contract No. AT(11-1)-1083.
The Depletion Effect in Pulse Polarography with Stationary Electrodes K. B. OLDHAM’ and E. P. PARRY North American Aviation Science Center, Thousand Oaks, Calif. Normal pulse polarography involves the application of voltage pulses of gradually increasing amplitude to an electrode which is held at a potential (resting potential) more positive than that a t which reduction occurs. The pulse duration i s a t least 50 times smaller than the time between pulses. With reversible systems reoxidation occurs during the time between pulses so that the reductant is fully regenerated. With irreversible systems, no reoxidation occurs, and a depletion effect is observed in unstirred solutions. This paper describes the depletion effect from both a theoretical and experimental point of view and points out its rather wide occurrence. Both stirring and proper choice of resting potential can eliminate this effect in applications of normal pulse polarography.
I
a succession of voltage pulses is applied to an electrode of constant area. The interval between pulses T exceeds the duration 7 of each pulse by a factor of 50 or greater. When employed analytically to study the electrode reaction Ox ne .-,Red (1) N NORMAL PULSE POL.4ROGRAPHY
+
the applied potential during the pulses steadily increases as shown in Figure Present address, Chemistry Department, The University, Xewcastle-uponTyne, England. 1
9 J 360
la, so that initially the potential is inadequate to cause the reduction of Ox. However, toward the end of the experiment the electrode is completely concentration polarized with respect to Ox during all pulses. Between pulses the electrode potential is returned to a resting value at which reaction (1) does not proceed from left to right. All of the theory and many of the experiments described in this communication are, however, concerned with one or other of the situations shown as b or c in Figure 1, in which the potential of the first pulse is already -ufficient to cause complete polarization with respect to Ox. The current measurement in pulse polarography is the difference between the instantaneous cell current measured toward the end of each potential pulse and immediately prior to the pulse. Measured currents are true diffusionlimited currents if the potential pulses are as in Figures 16 or IC. Normal pulse polarography can be used to advantage with solid electrodes. It has been shown, for instance, that in the reduction of silver ion a t a platinum electrode, the same curves can be obtained if the electrode is in stirred solution or is in a quiet solution (4). This is as expected since the silver is reduced for only the very brief period of the pulse but the electrode is kept for a much longer period at a potential where the silver metal is reoxidized so that no net deposit of metal accumulates on the electrode surface. Since the diffusion layer during pulses is thinner
than the shear layer in a stirred solution, stirring has essentially no effect on the polarogram obtained (4). The foregoing is completely valid if the resting potential is chosen so that the metal on the electrode surface is completely oxidized between pulses. However, in an unstirred solution and for a sufficiently irreversible system where a resting potential cannot be chosen a t which oxidation takes place B net depletion will occur during the very short pulse which will not be eliminated during the much longer waiting period between pulses. I n spite of the fact that the interval between pulses exceeds the pulse duration 50 to 100 ti,nei, this depletion effect is quite significant. The object of the present work is t’o describe this effect both experimentally and theoretically and show its wide applicability. The effect ha3 been qualitatively observed in the reduction of nickelous, chromate and iodate ions, and in the oxidation of ascorbic acid at the hanging mercury drop, in the reduction of gold(II1) at the pyrolyt’ic graphite electrode, and lead ion at t’he platinum electrode. Since adequate stirring eliminates the depletion effect, pulse polarographic analyses of irreversible systems should be carried out in the presence of stirring. THEORY
Prior to the first pulse, a uniform concentration C, of Ox exists throughout the cell. Subsequently, the Ox concentration a t a distance z from the VOL. 38, NO. 7, JUNE 1966
867
