The Rotating Disk Electrode Effect of Rates of Rotation and Polarization ILANA FRIED' and PHILIP J. ELVING The University o f Michigan, Ann Arbor, Mich.
The rotating disk electrode has been used to investigate the transition between the peak current and the limiting current regions in terms of the chief mode of mass transport which controls the electrode process. The oxidation of ferrocyanide in chloride solution on wax-impregnated graphite was the experimental system used. Experiments a t slow rotation speeds and fairly fast polarization rates show current peaks rather than plateaus, indicating that the main mode of mass transport of the electroactive species to the electrode surface is determined not only by whether the electrode is rotated or not, but also by the relative magnitudes of the speed of rotation and of the rate of polarization. Theoretical explanation of this phenomenon is given.
T
of using forced convection as a mode of mass transport to the surface of a n electrode was to increase the magnitude of the observed current and to ensure its stability, as compared to the current peaks observed in quiet solution; these objectives, the analytical importance of which is obvious, can be achieved by stirring the solution, rotating the electrode, or any other means of moving the solution past the electrode. The hydrodynamic situations with most forms of stirring are very involved, and it is frequently impossible to derive rigorous theoretical relationships. The most favorable situation is probably that of the rotating disk electrode, which has been the subject of many investigations, both theoretical and experimental. The reasons for this interest are two: the relative ease with which the electrode and the means of rotation are obtained, and the availability of a rigorous hydrodynamical theory for this configuration. The general situation in respect to the theory of the rotating disk electrode is well summarized in the standard reference by Levich ( 8 ) and by others ( I , 4 2 5 , 7 , 9, I O ) . Not all of these references are harmonious -e.g., one HE
ORIGINAL PURPOSE
Present address, Department of Inorganic and Analytical Chemistry, The Hebrew University, Jerusalem, Israel.
objective of Kolpanov ( 7 ) appears to be to prove that Levich is wrong in his treatment. The mathematical problem involved in the application to voltammetry is as follows
bC bt
=
D b2C - - v z bC bX2
bX
At time t = 0, at any x, and at any t in the bulk of the solution when x + m Co = Coo and
CR =
CR'
(2)
At the electrode surface
(3)
where C is the concentration of the electroactive species a t any point in the solution (the subscripts 0 and R refer to the oxidized and reduced forms, respectively; the superscript ' denotes bulk concentration), D is the diffusion coefficient of the electroactive species, v , is the component of velocity of the solution normal to the disk, i is the current passing through the cell, n is the number of electrons per molecule of electroactive species involved in the electrode reaction, F is the Faraday, A is the electrode area, R is the gas constant, T is the absolute temperature, and E and E" are the instantaneous and standard potentials, respectively. I n commonly encountered voltammetry, E is a function of time, usually linear, and i is a function of E. The solution of this problem has been attempted (3) and is not at all simple, Therefore, the present authors consider it useful for the understanding of the processes which control reversible reactions at rotating disk electrodes to present the following picture of the concentration gradient in the vicinity of a rotated disk electrode and the main mode of mass transport which controls the electrode process. For a stationary electrode, the rate of mass transport decreases with time, approaching zero asymptotically as the material farther from the electrode becomes progressively depleted. The observed current, therefore, diminishes with time. When the electrode is
rotated, however, the rate of mass transport cannot decrease below that maintained by forced convection. However, in the vicinity of the rotating disk, the component of velocity of the solution normal to the disk diminishes until it becomes zero at the surface of the disk; diffusion, therefore, gradually predominates over convection as the mode of mass transport of the electroactive species to the electrode surface. Thus, the concentration gradient, within a very short distance from the rotating disk electrode, closely resembles that a t a stationary electrode. The present study is part of a broad investigation of the theory, methodology, and applicability of graphite indicating electrodes, and is, consequently, specifically concerned with a graphite rotating disk electrode. EXPERIMENTAL
Materials. Solutions were freshly prepared b y weighing out and dissolving in distilled water a n appropriate amount of reagent grade potassium ferrocyanide iiliquots of such stock solutions mere diluted with solutions of reagent grade potassium chloride to form 1 m M K4Fe(CN)6 in 0.5M KC1 test solution. The graphite electrode was made of 0.125-inch ultrapurity spectrometric graphite electrodes (United Carbon Products Co.) impregnated with ceresin wax under vacuum ( 2 ) ; the side was coated with clear Krylon spray coating, which increased the diameter of the electrode slightly so that it fit more snugly into a 2.69-cm. diameter Lucite cylinder, out of which the electrode could still be pushed. Before each series of runs, approximately inch of electrode was removed by polishing, which was done by pushing the electrode inch out of the Lucite holder and then holding it against a rotating plate covered with No. 600 silicon carbide paper until the surface of the electrode and Lucite were flush. The resulting surface area reproducibility was good. Apparatus. The circuit used for application of the varying potential and measurement of the resulting current is given in Figure 1. The potential source was a HewlettPackard Model 2028 l o a frequency function generator, modified to produce only one cycle (6); the selected function was triangular, so that t h e VOL. 37, NO. 7, JUNE 1965
803
r\C
A
Figure 1 . Block diagram of circuit used for obtaining current-potential curves at the rotating disk electrode ( A ) Potential source (6) x-y Recorder ( C ) Electrolysis cell (R) Resistor box
potential changed linearly with time. Current-potential curves were recorded on a Mosely Autograf Model 135 x-y recorder at sensitivities of 0.1 volt/inch for the z axis and 0.5 mv./inch for the y axis. A Shallcross No. 819 decade resistance box was used. The circuit shown in Figure 1 was grounded as shown to bring the terminals of the function generator and the recorder to a common potential. The Lucite holder containing the electrode was slipped into a steel tube, which was rotated by a Cole-Parmer variable speed motor through a belt and a large (4.5-inch) pulley. The whole assembly was mounted on a heavy base sitting on the laboratory bench, which arrangement experiments indicated was sufficient to minimize vibration of the electrode. The resistance of the graphite indicating electrode with its mercury connection was generally less than 1 ohm, which would obviously not introduce a significant iR drop in the applied
Table I.
67
230
400
820
1 m M K4Fe(CN)e in 0.5M KCl;
polarization rate, 0.01 25 volt sec.-I
potential at the current values involved. The cell consisted of a crystallizing dish (8.9 cm. in diameter), a salt bridge with agar plugs at its ends, and a large saturated calomel electrode; the area of the latter (approximately 25.5 sq. cm.) was sufficient to minimize polarization. I n use, the dish contained about 100 ml. of test solution, in which the Lucite cylinder containing the electrode disk was immersed to a depth of about 0.4 em. Procedure. After resurfacing and removing graphite particles with tissue paper, t h e electrode holder was
ir
Polarization rate, volt see.+
il or zp, pa.
0.0124 0.0505 0.110 0.466 0,0124 0.0505 0,110 0.466 0.0124 0,0505 0.110 0.466 0,0124 0.0505 0,110 0.466 0.0124 0.0505 0.110 0.466
4.73c 9.03c 11.70. 21.80~ 7.77d 9,23c 12,o o c 22.10c 13,45$ 14.40d 13.90~ 23, OOc 17.90d 18.10d 18. l o d 24. OOc 25. 70d 26. 30d 26. 20d 26. 60d
190 195 203 213 202 203 212 227 216 203 209 233 227 213 208 247 242 - ~230 230
Ea14 - Ella,
EOb,
mv. 50 53 55 60 67 54 55 60 78 60 54 61 85 86 72 64 110
mv 140 142 143 147 153 152 153 155 153 153 151 149 157 153 148 147 157 152 148 146
111 108 76
.
Solution composition: 1mM K4Fe(CN)ein 0.5M KC1 solution. wave (obtained from intersection of the extrapolations of rising portion of wave and the residual current). c A current peak. E stated is E,,z. d A limiting-current region. E stated is ElIz. a
* Potential at foot of
804
ANALYTICAL CHEMISTRY
30
u'", rpm'" Figure 2. Variation of limiting current with square root of rotation speed
Effects of Rotation and Polarization Rates at the Rotating Disk Electrode"
Rotation speed, r.p.m. 0
20
IO
slipped into t h e steel tube. It then was dipped in a O.O30j, solution of Triton X-100 and immediately immersed in the test solution. RESULTS AND DISCUSSION
The experimental results are summarized in Table I and Figures 2 to 5. Figures 2 and 3 show the relationship between limiting or peak currents and rotation speed and polarization rate, respectively. The limiting current is proportional to the square root of the speed of rotation, as predicted by Levich and observed many times. The expected linear proportionality between peak current and the square root of polarization rate is not strictly observed. Inspection of Table I reveals that under some conditions a rotated electrode gives a current peak rather than a plateau. Current plateaus are obtained at low polarization rate; as rotation speed increases, the polarization rates that give plateaus become larger. This phenomenon suggests (a) that the controlling mode of mass transport depends not only on whether the electrode is rotated or not, but also on the duration of the experiment, and (b) that this last dependence changes with the rate of rotation. I n addition, under all conditions, the foot of the wave is a t the same potential (cf. right column of Table I). The latter potential was determined as the point of intersection on extrapolating the rising portion of the wave and the residual current. The changes in the values of half-wave potentials and of - Ell4)are, therefore, wave slopes not due to changes in the nature of the electrode process, but to other factors.
The half-wave or half-peak potential shifts to more positive potential as both polarizat'ion rate and speed of rotation increase. The magnitude of this shift, however, is much greater for the increase in speed of rotation for those curves that do not, have current peaks. The magnitude of shift of half-wave potential can, therefore, serve as an indication of the mode of mass transport' that is controlling the electrode process, as can also the change in the value of wave slope. On the basis of the criteria mentioned-(a) appearance or absence of a current peak, (b) value of half-wave pot,ential, and (c) magnitude of slolieone reaches the conclusion that the current in the experiments marked with a superscript c in Table I is convectioncontrolled and in those marked with a superscript d is mainly diffusion-controlled. Diffusion-controlled processes yield approximately the same peak current regardless of rate of rotation of the electrode (Figure 4). On the other hand, convection-controlled processes give the same values of limiting current regardless of rate of polarization (Figure 5 ) . These results can be explained both qualitatively and quantitatively. The qualit,ative explanation runs as follows. A concentration gradient starts to form in the solution close to the electrode surface as soon as the electrode reaction begins. As the experiment proceeds, more and more of the electroactive material reacts a t the electrode surface and more is transported from the s o h tion to the electrode surface. Thus, the distance from the electrode surface, a t which the concentration differs from that in the bulk of the solution, becomes progressively greater. Since mass transport very close to the electrode surface is mainly by diffusion, the concentration gradient at this distance resembles that a t a stationary electrode. When the time of the experiment is not sufficient for the concentration gradient to develop to the distance where both convection and diffusion play equal roles in mass transport, or when convection predominates over diffusion, the currentvoltage curve resembles that obtained a t stationary electrodes. Quantitatively, the peak current is given for diffusion-controlled processes by the Randles-Sevcik equation: id
=
/
0
-
/
/
-B'
a. a
.-
10
0 /
0 I
I
I
1
0.1 0.3 0,5 0.7 Square Root of Polarization Rate, ( v. Sec'')'/2 Figure 3.
Variation of peak current with square root of polarization rate
1mM KaFe(CN)s in 0.5M KCI; X, quiet solution;
0,67 r.p.m.;
+, 230 r.p.m.;
0400 r.p.rn.
I A
-
1.3I
.-
P
-+
+
+
/Q-
0
L
n
.-
L.-
l
o
- 2.0
- 1.5
- 0.5
-1.0 Log ( polarization rate )
Figure 4.
Variation of limiting or peak current with speed of rotatioli
1mM K$e(CN)G in 0.5M KCI; X, 0.0124 volt ret.-';
0, 0.466 volt ret.-'
0,0.0505
volt ret.-';
+, 0.1 10 volt set.-';
0.454 nl.'.lCo (l>anF/RT)1'2( 5 )
where i d is the diffusion current and a is the rate of polarization. For convection-controlled processes the limiting current is given by the Levich equation :
it
/ 20
I
I.78
2.0
nFADw'"T0 '1,61D"3~1'~ (6)
where v is the kinematic viscosity of the solution and w is the rot,ation speed. When there is a transition from dif-
I
I
2.3
2.7
Log w Figure 5.
Variation of limiting or peak current with polarizathn rate
1mM KIFe(CN)e in 0.5M KCI; X, quiet solution; 820 r.p.m.
3 ,67 r.p.m.; f, 230 r.p.m.;
0, 400 r.p.m.; A,
VOL. 37, NO. 7, JUNE 1965
805
Table II.
Rotation speed r.p.m. 67
230
400
820
See text.
Predicted Mode of Mass Transport Under Specified Conditions
Polarization rate volt see. 0.0124 0.0505 0.110 0.466 0.0124 0.0505 0,110 0.466 0.0124 0.0505 0.110 0.466 0.0124 0.0505 0.110 0.466 See text.
Calcd. i ~ , Ira. 7.31
13.6
17.9
25.6
fusion- to convection-controlled processes, the larger current of the two will be observed. The following values were used in the calculations for 2 5 O , which are summarized in Table 11: n = 1, F = 96,500 coulombs eq.-l, A = 0.0794 om2, D = 7.4 x sq. cm. sec.-l, Co = 10” mole ern.+, sq. and Y = 0.0789 cm./sec. Comparison of Tables I and I1 shows
Calcd. id, Ira. 6.56 13.3 19.4 42.1 6.56 13.3 19.4 42.1 6.56 13.3 19.4 42.1 6.56 13.3 19.4 42.1
Predicted control Convection Diffusion Diffusion Diffusion Convection Convection Diffusion Diffusion Convection Convection Diff usiona Diffusion Convection Convection Convection Diff usionb
an agreement between the predicted and observed modes of mass transport except for the two cases marked with superscripts a and b. I n the case marked a, ir and i d are close in value. I n the case marked b, other factors may be involved, such as slight irreversibility of the electrode reaction, as might also be indicated by the abnormally high wave slopes.
The present study is apparently the first one concerning the range of transition between diff usion-controlled and convection-controlled processes. The transition is quite sharp in spite of what might be intuitively expected. LITERATURE CITED
(1) Bowers, R. C., Ward, G., Wilson, C. AI., DeFord, D. D., J . Phys. Chem.
65,672 (1961).
( 2 ) Elvine. P. J.. Smith. D. L.. ANAL.
CHEM.52,1849 (1960). ’ (3) Fried, I., Elving, P. J., Ibid., 37, 464 (1965). (4) Gregory, D. P., Riddiford, A. C., J . Chem. SOC.1956, p. 3756. (5) Hale, J. AI.! J . Electroanal. Chem. 6 , 187 (19631. (6) Hewlett-Packard Co., Palo Alto, Calif., Application Note 31. (7) Kolpanov, P. L., Zhur. Fiz. Kim. 35, 1538 (1961). (8) Levich, V. G., “Physicochemical Hydrodynamics,” p. 65 et seq. PrenticeHall, Englewood Cliffs, N.J., 1962. (9) Siver, Yu. G., Russian J . Phys. Chem. 33,533 (1959). (10) Ibid., 34, 273 (1960). RECEIVEDfor review March 9, 1964. Resubmitted September 28, 1964. Accepted April 19, 1965. Work supported in,part by the U.S.Atomic Energy Commission.
Determination of Alkylaluminum Compounds by Thermometric Titration W. L. EVERSON and EVELYN M. RAMIREZ Shell Development Co., Emeryville, Calif.
b Aluminum alkyls, alkylaluminum hydrides, and alkylaluminum halides can be analyzed simply, rapidly, and accurately by thermometric titration. Conditions have been found under which ketones are specific titrants for alkylaluminum hydrides, the exploratory studies of Hoffmann and Tornau have been extended to small samples, and a “cleanup” technique eliminates the effects of traces of water and oxygen in the reaction system. Selection of titrants permits determination of R3AI- and R3AIH-type compounds separately, or in sum. Alkylaluminum halides behave much as R3AIcompounds. Alkoxide (R2AIOR) can b e estimated thermometrically, but is better determined independently or as the difference between total and active aluminum. Thermometric titration is applicable to all other alkylaluminum compounds tested in the C I - C ~range.
N
methods have been described for the determination of alkylaluminum compounds. Bonitz ( 2 ) studied their reactions with isoquinoline UMEROUS
806 *
ANALYTICAL CHEMISTRY
by conductometric and potentiometric titration and described the strongly colored complex formed with diethylaluminum hydride. Farina, Donati, and Ragazzini (8) and Nebbia and Pagani (16) have reported modified potentiometric titration procedures. Neumann (17 ) studied the colored complexes of alkylaluminum hydrides in more detail, and also developed a gasometric method for hydride based on reaction with N methylaniline a t low temperatures. hlitchen (15) extended the isoquinoline color reaction to the siniultaneous spectrophotometric determination of RaAl and R2A1H,and Wadelin (19) used it as the basis for photometric titration to determine activity. Razuvaev and Graevskii (18) developed a visual indicator titration; the chemistry of reversible color-indicator reactions was discussed by Hagen and Leslie (10). Bartkiewicz and Robinson ( 1 ) described an iodimetric method. Ziegler and Gellert (21) reacted the sample with ammonia and, after alcoholysis of the amides formed, determined the liberated ammonia.
Several methods involve decomposition and analysis of the resulting gases. Lioznova and Genusov (14) used simple hydrolysis and measurement of the gas liberated; more refined methods have employed mass spectrometric analysis [Keumann (17) and Ford and Hagen ( 9 ) ] or gas chromatography [Crompton and Reid ( 4 ) and Dijkstra and Dahmen ( 5 ) ] . Infrared data, indicating the possibility of determining Et2AlH and Et2A10Et in &AI, were published by Hudson (13). Crompton (3)reported a direct method for alkoxide based on low-temperature reaction with acetic acid and determination of the alcohol liberated. Hoffmann and Tornau described dielectric constant ( DK) titration (11 ) and thermometric (calorimetric) titration (18) of alkylaluminum compounds. Using D K titration, they verified Neumann’s conclusion that, in reaction between an alkylaluminum hydride and the azomethine group (-CH=XR) , the first step is reduction by the hydride to form an amide and the second step is reaction of the amide with a second azomethine group to form an electron-
