IRVING
SHAIN AXD KENXETH J. MARTIN
Vol. 65
explanation;atit,he present time. There is, however, the two cases. There is, however, considerable the difficulty that if the hybridization of the nitrogen evidence for such changes in hybridization, so that changes with the back coordination the resulting possibly such an unfavorable energy requirement resonance form would not be very favorable, since is more than compensated for by the formation of the nuclear spatial structure would be different in the p,-d, bond.
ELECTROLYSIS WITH CONSTANT POTEKTIAL: REVERSIBLE PROCESSES A T A HANGING MERCURY DROP ELECTRODE1 BY IRVING SHAINAND KENNETH J. MARTIN Department of Chemistry, University of Wisconsin, Madison, Wisconsin Received July 6,1060
The technique of electrolysis with constant potential using a stationary spherical mercury electrode (the hanging mercury drop) was investigated under conditions where the curvature of the electrode surface had to be considered. Current-time curves, covering periods from 1 to 25 seconds after the start of the electrolysis, were compared with theory for several cases. It was found that the results agreed with the classical equations for reactions at potentials where the current was controlled by diffusion alone. In addition, equations for reversible processes taking place a t potentials near the equilibrium potential were verified for reactions in which both the reactant and product were soluble in the solution. For systems in which the product of the reaction formed an amalgam, the convergent nature of the diffusion processes within the hanging mercury drop electrode had to be considered. The general suitability of the hanging mercury drop electrode for potentiostatic studies was also investigated, and the effects of shielding of the electrode and convection were characterized.
I n recent years, the hanging mercury drop elec- Under these conditions, the curvature of the electrode has proved to be useful in a variety of electro- trode must be considered when using the hanging chemical applications. Since first reported by mercury drop electrode. Equations describing some of the current-timeGerischer, the electrode has been used for kinetic studies, including galvanostatic, potenti~static,~potential relations a t a spherical electrode for elecand linearly varying potential methods.b In addi- trolysis with constant potential have been known tion, it has been applied in a useful series of analyti- for many years. However, until recently, a suitcal methods based on voltammetry with linearly able stationary spherical mercury electrode has not varying potentiaL6 This stationary spherical elec- been available for rigorous verification of the equatrode is convenient t o use, has an exactly reproduc- tions. This paper describes a study of such ible size and shape, and its geometry is such that potentiostatic processes for reactions not comequations are readily obtainable for processes plicated by kinetic effects. I n addition, the limitainvolving diffusion to the electrode surface. This tions of the hanging mercury drop electrode and is particularly true in the case of potentiostatic the sources of error involved were studied. D8usion Controlled Limiting Currents.-If the processes, since application of the Laplace transform frequently leads directly to a useful solution of potentiostatic reduction of substance 0 to substance R is considered a t potentials sufficiently the boundary value problem involved. WIn all of the previous applications of the hanging cathodic that the current is controlled entirely by mercury drop electrode to potentiostatic studies, t,he diffusion of 0 to a spherical electrode surface, the curvature of the electrode surface could be the equation for the current-time curve is7 ignored, Le., the experiments were completed in such short times (microsecond to millisecond range) that the corresponding equations for diffusion to a plane electrode could be applied. However, there where i is the current flowing, n is the number of are many cases in which the accuracy and con- electrons involved in the electrode reaction, F is venience of more conventional equipment, including the faraday, A is the area of the electrode, DO is pen-and-ink recorders, is required. In such cases, the diffusion coefficient of substance 0, C*O is the data cannot be obtained a t times much shorter bulk concentration of substance 0, tis the time after than 1 to 2 seconds after the start of electrolysis. the sudden application of the constant potential, and is the radius of the electrode. The corre(1) Based, in part, on the Ph.D. thesis of Kenneth J. Martin, Unisponding equation for diffusion to a plane electrode versity of Wisconsin, 1960. (2) H. Gerischer, 2. physzk. Chem., 202, 302 (1953). lacks the term in T O , and when T O is large, or when t is (3) T. Berains and P. Delahay, J . A m . Chem. Soc., 77, G448 (1955); small, the contribution of this term to the total H. Mateuda, S. Oka and P. Delahay, zbid., 81, 5077 (1959). current is small. For the experimental conditions (4) H. Gerischer and W. Vielstich, Z . piwsik. Chem., N.F., 3, used in this work, however, this contribution was as 16 (1955); W. Vielstich and €1. Gerischer, ibrd., 4, 10 (1955); H. Gerischer and K. Staubaoh, zbid., 6, 118 (1956). much as 30’% of the total current. ( 5 ) R. D. DeMars and I. S h a h J . Am. Chem. SOC.,81, 2654 (1959). Attempts by Laitinen and Kolthoff8 to verify (6) (a) J. W. Ross, R. D. DeMars and I. Shain, Anal. Chem., 28, 1768 (1956); (b) R. D. DeMars and I. Shain, ibid., 29, 1825 (1957); (c) K. J. Martin and I. Shain, ibid., SO, 1808 (1958); (d) W. Kemula and Z. Kublik, Anal. Chim. Acta, 18, 104 (1958): (e) TT-. Kemula. Z . Kuhlik and S. Qlodoaski, J. Electroanal. Chem., 1, 91 (1959).
(7) P. Delahay, “New Instrumental Methods In Electrochemistr) ,” Interscience Publishers, New York. N. Y.,1954, B. 61. (8) H. A. Laitinen and I. M. Kolthoff, J . A m . Chem. Soc , 61, 3344 (1939).
Feb., 1961.
ELECTROLYSIS WITH COXSTANT POTENTIAL
equation 1 at a, spherical platinum electrode were unsuccessful because of convection encountered a t the relatively long electrolysis times they used. Skobets and KavetskiiDused equation 1 to determine diffusion coefficients at a stationary spherical platinum electrode. The current can be plotted as a funct,ion of (l/t)l’z and the slope of the resulting straight line can be used to calculate the diffusion coefficient. Equation 1 was used in this work as a standard for evaluation of the characteristics of hanging mercury drop electrodes, as well as for the determination of several diffusion coefficients. An analysis of the current-time curves obtained for diffusion controlled limiting currents a t hanging mercury drop electrodes is shown in Fig. 1 (closed circles). A solution containing 5.19 X 10-4 M thallous ion, 1 M potassium nitrate and 0.1 M acetic acid-sodium acetate buffer was electrolyzed at -0.650 v. (us. S.C.E.), using hanging drop electrodes formed bly collecting 1, 2 or 3 drops of mercury from the capillary. Fairly large deviations from the theoretical straight lines were observed. The two major sources of error were found to be convection (currents higher than theory) and shielding of the electrode (currents lower than theory). Influence of Convection.-In all of the systems studied, interference caused by convection was observed at both very short times, and a t relatively long times. The convection a t long times was found to be caused by vibration and by density gradients in the cell, and could be noted as abnormally high and unreproducible currents. By carefully isolating the cell from sources of vibration, and by allowing the solution to come to rest for several minutes after performing any manipulations on the cell, it WBS possible to carry out electrolyses for about 30 seconds wit’houtinterference from these sources of convection. Convection a-t very short times was found to be caused by movement of the hanging mercury drop electrode. On sudden application of the working potential, the surface tension of the mercury changes, resu1ti:ng in a slight change in the shape of the mercury electrode. Such movement of the electrode is observed easily with a low power microscope. The effect of this movement of the electrode, and the resulting convection in the solution, is shown in Fig. 1. The high current caused by stirring of the solution is more pronounced with the heavier electrodes, and persists for as long as 4 seconds. X o attempts were made to determine when effects of drop movement are first noticeable. However, some recent results obtained by Delahay and Oka’O indicate that these effects may be present a t times as short as 1millisecond. In an attempt to minimize the effect.s of movement of t,he mercury, an electrode was construct’ed which supported the mercury drop from below. Such an electrode showed considerably less convection a t short tiines (Fig. 1, open circles). The 3 drop electrode could not be used in this case, how(9) E. 31. Skobnts a n d N. 9. Kavetakii, Zhur. Fiz. Khim., 24, 1486
(1950). (10)
P De1:rIiny a n d P. O k x . .I. A m . Chem. Soc., 82, 329
(1960).
255
t, sec.
25
12
6
3
I
I
2 I
1.5
I 2 1.0
I
I
I 0
’[ 8
I
0.2
I
0.4
1
I
0.6
I
I
0.8
I
I
I .o
i/t”z Fig. 1.-Analysis of currentrtime curves for the reduction of 5.19 X M thallous ion, a t -0.650 v. (os. S.C.E.): A, 1 drop electrode, r0 = 0.0515 cm.; B, 2 drop electrode, r0 = 0.0650 cm.; C, 3 drop electrode, T~ = 0.0744 cm. Lines, theoretical, based on Do = 1.79 X loe5 cm.*/sec.; 0, hanging mercury drop electrode; 0, supported mercury drop electrode.
2
10.1 CM.1 Fig. 2.-Tip of hanging mercury drop electrode, comparing shielding for 1 and 2 drop electrodes.
ever, because the weight of the mercury caused the electrode to flatten out into a distinct,ly non-spherical shape. Influence of Shielding.-The means by which the hanging mercury drop electrode is supported interferes to some extent with the mass transfer process to the electrode surface. The actual point of contact of the mercury with the platinum wire on which the drop is hung is only a very small portion of the total area of the electrode (less than 0.4% in most cases) and the main interference is caused by the distortion of the diffusion layer by the glass. This
IRVING SHAINAND KENNETH J. MARTIN
256
/ I
\\
I
I
I
\
I
\
/
I
0.1 CM n Fig. 3.-Calculated diffusion layer (one per cent. depletion) for a hanging mercury drop electrode: Do = 2 x 10” cm.2/sec., t = 10 8ec., ro = 0.0515 cm.
pr SEC.
I
l 0 P + + I
0.2
I
I
I
I
0.4
0.6 I/t ’’2
0.8
1.0
Fig. 4.-Analysis of current-time curves for the reduction of 1.07 X 10-8 M titanium(1V) oxalato complex: experimental; lines, theoretical; TO = 0.0678 cm. plied potentials were A, -0.450 v.; B, -0.330 v.; C, -0.310 v.; I>, -0.200 v.; E, -0.270 v.; F, -0.250 v.; each us. S.C.E. ~~~~~
effect can be minimized by drawing out the glass tip during construction of the electrode. An approximate scale drawing of the tip of the electrode (Fig. 2) indicates that shielding is relatively severe near the top of the mercury drop, but is decreased as larger electrodes are formed by collecting additional drops of mercury from the capillary. Thus, shielding should result in currents lower than predicted by equation 1, and the effect should decrease as the size of the hanging mercury drop electrode is increased. The results (Fig. 1) are surprising since the deviation in the current is apparently independent of time, at least for times greater than about 4 seconds, when convection due to drop movement ceases. The effect is more severe with a mercury drop electrode supported
VOl. 65
from below since considerably more glass must be retained in order to have a surface large enough to collect the drops of mercury. However, even with the supported mercury drop electrode, shielding is independent of time for the conditions of these experiments. Since the radius is the only term in equation 1 which can shift the curve vertically without affecting its slope, the effective radius of the electrode must differ from the geometric radius. That this could occur easily is shown in Fig. 3, since a rapid thickening of the diffusion layer where the mercury approaches the glass could cause the diffusion layer to become hemispherical. The dotted line in Fig. 3 represents the thickness of the diffusion layer (1% depletion) 10 seconds after the start of electrolysis, basing the calculations on equation 1. As a result of these studies, it is apparent that the use of the hanging mercury drop electrode for potentiostatic investigations involves a compromise between the effects of shielding, and convection caused by drop movement. Interference caused by convection decreases as the size of the electrode decreases, but the effects of shielding are most severe with the small electrodes. The shielding was so severe with supported mercury drop electrodes that they were not considered useful. For this work, hanging mercury drop electrodes formed by collecting 2 drops of mercury from the capillary were used. With these electrodes (TO = 0.06 cm.) currents measured between about 4 and 25 seconds were not affected by either convection or shielding. Determination of Diffusion Coefficients.-In the couwe of this work, it was possible to evaluate the errors involved in the use of equation 1 for determination of diffusion coefficients. Diffusion coefficients obtained from both the slope and the intercept of the i vs. (l/t)1’2 plots agreed closely. The values are accurate to at least *2%, indicating that this method should be very useful for the accurate determination of diffusion coefficients. The diffusion coefficient obtained for thallous ion in 1 M potassium nitrate, 0 . 1 M acetic acid and 0.1 AI sodium acetate was 1.79 X cm.2/sec. This agrees exactly with the value in 1 M potassium nitrate reported by Rulfs.” The diffusion coefficient of the titanium(1V) oxalato complex in 0.2 ill oxalic acid, and 1% sulfuric acid ( p H 1.25) was 0.61 X cm.2/sec., a value somewhat lower than that calculated from the diffusion current constants given by Pecsok.12 The diffusion coefficient of iodate ion in 0.1 M potassium nitrate, 0.1 M potassium dihydrogen phosphate, and 0.1 AI potassium monohydrogen phosphate ( p H 7.2), was 1.01 X cm.?/sec., compared with the infinite dilution value of 1.09 X ~m.~/sec.1~ Reversible Electrode Reactions : Reactant and Product Soluble in the Solution.-If the applied conqtant potential is not restricted to very cathodic values as above, the current flowing at the electrode is controlled by potential as well as mass transfer. The equation describing the current-time curves (11) C. L. Rulfs, J A m Chem S o c , 7 6 , 2071 (1054). (12) R. L. Pecsok, rbzd., 73, 1304 (1951). (13) I. M. Kolthoff and J. J. Lingane. “Polarography,” 2nd ed.. Interscience Publishers, New York, N. Y.. 1952, Vol. I, p. 52.
ELECTROLYSIS WITH
Feb., 1961
257
CONSTANT P O T E K T I S L
for the case where substance R is initially absent is
where y =: ( D o , ~ D R ) 'and /~, e
=
c o = exp [ n m F CR
(-~EO)]
(at r = ro)
(3)
Dp,is the diffusion coefficient of substance 11, Co and CR are the concentrations of substances 0 and R , respectively, r is the distance from the center of the electrode, 0 defines the ratio CO/CR a t the electrode surface in terms of the Nernst equation, and the other terms were defined previously. Although equation 2 apparently has not been reported previously, its derivation follows readily from the corresponding case involving a plane electrode. l4 Again, the only difference between the two c;zses is that the equation for the plane electrode lacks the term in ro. At very cathodic potentials, e + 0, and equation 2 approaches equation l. I n order to verify equation 2, the reduction of the titanium(1V) oxalato complex was selected on the basis of its reported reversibility,12 and because the products of the electrode reaction are soluble in the solution. The current was plotted as a function of (1,'t)''~ at various potentials (Fig. 4). By assuming that y = 1, an effective Eo of -0.287 i 0.001 v. vs. S.C.E. mas calculated from the slopes of the lines. The agreement between theory and experiment is excellent, except a t short times (less than 4 seconds) where convection caused by drop movement was severe. Reversible Electrode Reactions : Products Soluble in the Mercury.-In attempting to apply equation 2 to the analysis of current-time curves obtained for the reduction of thallous ion, major deviations were observed (Fig. 5 ) . Curve A was recorded a t a potential sufficiently cathodic that a diffusion controlled limiting current was obtained ; the agreement with equation 1 was satisfactory. At least cathodic potentials, the current was less than that predicted by equation 2, and the deviation mas apparently independent of time of elect rolysis. -4 re-examination of the boundary conditions used in the derivation of equation 2 revealed that the difficulty probably was connected with the convergent nature of the diffusion layer within the mercury drop, and also by the fact that the concentration of substctnce R builds up within the limited volume of the dectrode during the course of the electrolysis. Any attempt to account for both these effects rigorously appeared extremely complex. However a consideration of the convergent nature of the diffusion layer within the mercury leads to some useful conclusions. Theory.--The boundary value problem requires solving F'ick's second law for spherical diffusion'j (4) (14) Kef. 7, I , , 52 57 (15) Ref. 7, p . fill.
e
'
0.2
0.3
e
0.4 I /t
0.5
0.6
Fig. 5.-Analysis of current-time curves for the reduction of 5.19 X M thallous ion: points, experimental; lines, calculated from equation 2 . The applied potentials were: A, -0.650 v.; 13, -0.530 v.; C, -0.480 v.; each v s .
S.C.E.
c
3
/
I
I
I
I
I
I
-A5
-.50
-35
-.60
-65
VOLTS VS. SCE . Fig. 6.-Current flowing 10 sec. after start of electrolysis for the reduction of 5.19 X M thallous ion: points, experimental; lines, theoretical-calculated from: A, equntion 2; B, equation for diffusion to a plane electrode; C equation 14. (5)
The initial conditions are Co = c o * ; CR = 0, for t = 0, any r
(6)
The boundary conditions are CO+
CR+O
CO* as as
r+ r+
a, any t 0, a n y t
(i) (8)
Equation 8 is valid only until the diffusion layer of substance R reaches the cent,er of the electrode, since the flux of R is zero a t r = 0. An additional boundary condition reflects the fact that when an amalgam is formed, the flux and direction of diffusion are the same for both reactant and product
IRVING SHAIXAXUDKENNETH J. MARTIX
258
1-01. 65
Furthermore it is assumed that the Nernst equation holds (equation 3). Although an exact analytical solution to this problem was not obtained by application of the Laplace transformation because of the complexity of the transform of the solution, an approximate solution was obtained by assuming
rent vs. (l/’t)’/~gave straight lines parallel to, but higher than, the theoretical curves predicted by equation 14, except a t very anodic potentials where the currents were very low. The effect on the current a t various potentials can be seen in Fig. 6 as the controlling diffusion process shifts from the mercury to the solution. The current flowing a t t = 10 seconds is plotted as a function of potential co + C R = co* (10) from equations 2 and 14; and for comparison purEquation 10 can be derived from geometric coli- poses, the curve obtained from the corresponding siderations involving diffusion to a plane electrode. l6 equation for a plane electrode is also included. It It is also valid for diffusion to a sphere if the prod- is interesting to note that the potential a t which the ucts of the electrode reaction are soluble in the solu- current flowing happens to cross the curve for the tion, and if it is further assumed that Do = DR. plane electrode is the calculated effective EO. For the case of amalgam formation, equation 10 Experimental is only an approximation which holds best when CR Apparatus.--4 potentiostat used in this work. The is low, Le., when relatively low currents are ob- circuit was arranged so thatwas the potential between the served a t less cathodic potentials. working electrode and a reference electrode was controlled to Combining equations 10 and 3, an alternate within 1 millivolt, while the current through the cell passed between the working electrode and a counter electrode. boundary condition is obtained CR
=
CO*/(1
+ 6) fort > 0,
T
=
TO
(11)
A solution to this boundary value problem is’?
By differentiating equation 12 with respect t.o r, and evaluating a t r = ro, a value of the flux is obt,ained, which when substituted into Fick’s first law yields
For the experimental conditions encountered in this work, t,he exponential terms can be neglected, and equation 13 reduces to
This is the same as equation 2 except that the term in r0 is negative. Thus the convergent nature of diffusion within the electrode apparently affects that contribution to the total current represented in the term in TO. Equation 14 should be valid a t the less cathodic values (large e), and equation 2 (or 1) should hold a t small values of 6. Results and Discussion Current-time curves for the reduction of thallous ion were obtained a t various potentials in order to test some of the above conclusions. Plots of cur(16) Ref. 7, D. 54.
(17) J. Crank, “The Matheniatics of Diffusion,” Oxford University Press, London, 1956, p. 86; H.S. Carslan and J. C. Jaeger, “Conduction of Heat in Solids,” 2nd ed., Oxford University Press, London, 1959 p. 233.
The instrument was based on the operational features of the analog computer amplifiers manufactured by G. -4.Philbrick Researches, Inc. (Boston), and used some of the ideas suggested by DeFord.18 The current-time curves were recorded on a Leeds and Northrup Speedomax recorder, 1 second response, 1 millivolt full scale. The hanging drop electrode and the electrode assembly were essentially the same as described previously.& The counter electrode was a platinum foil connected to the cell by means of a 2 M potassium nitrate salt bridge terminated with a 10 mm. ultrafine sintered glass disc. The reference electrode was a Beckman calomel electrode, placed in a compartment containing 2 M potassium nitrate, and connected to the cell by means of a Luggin capillary salt bridge. For those experiments in which the mercury drop was supported from below, an electrode was constructed by sealing platinum wire (0.016 inch diameter) into the end of a 6 mm. soft glass tube. The end of the electrode was bent in the shape of a J and then polished slightly concave so that drops of mercury falling from a capillary could be collected. An effective method of isolating the cell from vibration was devised by mounting the entire cell assembly, waterbath, and a heavy slate block on an inflated inner tube. All experiments were carried out a t 25.0’ in a thermostat. Materials .-Purified K2TiO(CzOa)~2H20was recrystallized once, and analyzed by precipitation with cupferron. The thallium solution was analyzed by precipitation with chromate. All other materials were reagent grade, and were used without further purification. To remove oxygen from the cell, Linde high purity nitrogen was used without further purification.
Acknowledgments.-This work was supported in part by funds received from the United States htomic Energy Commission, under Contract No. -4T(11-1)-64, Project KO.17. Other support mas received from the Research Committee of the Graduate School of the Tiniversity of Wisconsin with funds received from the Wisconsin Alumni Research Foundation. Some of the instruments used in this work were purchased with funds made available by the Standard Oil Foundation, Inc. (Indiana). (18) D.D.DeFord, Division of Analytical Chemistry, 133rd Meeting. A.C.B., San Francisco, Cahfornia, April, 1958.
