J . Phys. Chem. 1990, 94, 6506-6509
6506
until it was placed in the Mossbauer cryostat. P-Hydroquinone Xenon Clathrate. The reaction vessel for the preparation of the @-hydroquinonexenon clathrate consisted of a thick walled glass tube with a side arm having a total volume of 100 mL. After placing 3.0 g of hydroquinone (C6H4(OH),) and 30 mL of distilled water in the tube, it was evacuated and enough xenon (59 mmol, 7.7 g) condensed into the side arm to yield a pressure of 21 atm at ambient temperature. The reaction vessel was heated to 70 OC on a water bath to dissolve all the hydroquinone and then allowed to cool in a well-insulated Dewar to ca. 30 OC over a period of 24 h, with occasional shaking. The vessel was then maintained at room temperature for another 24 h to allow the system to reach equilibrium. After cooling to 0 OC, excess xenon was recondensed into the side arm, which was then heat-sealed and detached from the vessel. After filtering and air-drying, the product was mixed with CCI, and allowed to settle according to a procedure described by Powell." Approximately I % of the total material floated to the surface and was decanted with the CCI,. The remaining solid was again air-dried, the needlelike crystals ground and samples analyzed for xenon content. After liberating the xenon by dissolving a weighed sample in anhydrous diethyl ether, followed by air-drying, the calculated composition was 26.1 wt % xenon and 73.9 wt % clathrate, corresponding to 92% occupancy of the
available cavities based on the limiting formula 3C6H4(OH)2-Xe. Before the product was ground, a few crystals were selected by hand and used to determine the crystal structure of the @hydroquinone xenon clathrate.l2 A ground sample was transferred to a 20.0-mm-i.d. Kel-F Mossbauer sample holder. Mossbauer Spectroscopy. The source of 57.6-keV radiation for 1271 Mossbauer spectroscopy was Mg3127mTe06.Details of the Iz7I source preparation and spectrometer used to obtain natural abundance Miissbauer spectra on IBXe and IZ7Isamples have been described el~ewhere.~ Both sources and absorbers were immersed in liquid helium (4.2 K) in a research cryostat manufactured by the Janis Research Corp. The velocity scale was calibrated by using a standard iron foil and a 57Co/Rh source mounted on the reverse end of the transducer; calibration spectra were thus recorded without interruption of the drive sequence. The calibration spectra were computer-fitted to give a linear velocity scale and folding point. Folded spectra were fitted by using the program G M F P I ~ which incorporates full transmission integral procedures. Raman Spectroscopy. Details of the Raman instrumentation have been described elsewhere.], The spectra were recorded in sealed 6.5-mm-0.d. X 4.0-mm-i.d. Pyrex glass tubes at ambient temperature. (13) Ruebenbauer, K.; Birchall, T. Hyperfine Inreracr. 1979, 7, 125. (14) Syvret. R. G.; Schrobilgen, G. J. Inorg. Chem. 1989, 28, 1564.
Oscillations during the Electrocatalytic Reduction of Hydrogen Peroxide on a Platinum Electrode N. Fetner and J. L. Hudson* Department of Chemical Engineering, Thornton Hall, University of Virginia, Charlottesoille, Virginia 22903-2442 (Received: January 9, 1990; In Final Form: March 23, 1990)
The galvanostatic reduction of hydrogen peroxide on a platinum rotating disk electrode was studied in acidic solutions. The nature of the oscillatory response in potential is described as a function of hydrogen peroxide concentration, disk rotation rate, and applied current.
Introduction Oscillatory behavior during electrochemical reactions has been known for more than 150 years1 and has been investigated by many In the past few years detailed dynamic studies have been made on electrochemical reactions. Time series of current or potential have been analyzed by methods of nonlinear dynamics; phenomena such as bifurcations from steady to oscillatory states, quasiperiodicity, and various types of low-order chaos have been explored. Many of these studies have been carried out with electrodissolution of metals such as copper, iron, and nickel, usually in acidic Oscillations have also been observed during electrocatalytic reactions including the oxidation of hydrogen,I"l9 formaldehyde and formic acid,20s21methanol,21and organic fuelsz2as well as the reduction of nitric acid2' and hydrogen In many of these studies the reactions were carried out on an electrode such as platinum in a supporting electrolyte such as HZSO,, HCIO,, or HCI. Schell et studied the dynamics of two electrocatalytic reactions; they examined the details of the oscillatory states occurring during the electrochemical oxidation of formaldehyde and formate/formic acid. They found interesting dynamics such as a periodic-chaotic sequence and a sequence of Farey states. Thus, *To whom correspondence should be addressed. 0022-3654/90/2094-6506$02.50/0
it is seen that electrocatalytic reactions can under some circumstances exhibit interesting behavior complementary to that seen ( 1 ) Fechner, G . Th. J . Chem. Phys. 1928, 53, 129.
(2) Wojtowicz, J. In Modern Aspects ofElectrochemisrry; Bockris, J . O., Conway, B., Eds.; Plenum Press: New York, 1972; Vol. 8. (3) Copper, J.; Muller, R.; Tobias, C. J . Elecrrochem. SOC.1980, 128, 1733-1744. (4) Lee, H. P.; Nobe, K.; Pearlstein, A. J . Elecrrochem. Soc 1985, 132, 103 1-1 037. (5) Lee, H. P.; Nobe, K. J . Elecrrochem. SOC.1986, 133, 2035. (6) Pearlstein, A. J.; Lee, H. P.; Nobe, K. J . Electrochem. SOC.1985, 132, 2159. (7) Tsitsopoulos, L. T.; Tsotsis, T. T.; Webster, I. A. Sur5 Sci. 1987, 191, 225-238. (8) Diem, C. B.; Hudson, J . L. AIChE J . 1987, 33, 218-224. (9) Albahadily, F. N.;Schell, M. J. J . Chem. Phys. 1988, 88, 4312-4319. (IO) Bassett, M. R.; Hudson, J. L. J . Phys. Chem. 1988,92,6963-6966. (1 1) Lev, 0.;Wolfberg, A,; Sheintuch, M.; Pismen, L. M. Chem. Eng. Sci. 1988, 43, 1339-1353. (12) Bassett, M. R.; Hudson, J. L. J. Phys. Chem. 1989,93,2731-2737. (13) Bassett, M. R.; Hudson, J. L. Physica 1989,035, 289-298. (14) Butler, J. A. V.; Armstrong, G. Narure 1932, 129, 613-614. (15) Armstrong, G.; Butler, J. A. V. Discuss. Faraday SOC.1947, No. I , 122-126. (16) Horanyi, G.; Visy, Cs. J . Elecrroanal. Chem. 1979. 103, 353-361. (17) Kodera, T.; Yamazaki, T.; Kubota, N . Elecrrochim. Acra 1986, 31 ( 1 1 ) . 1477-1478. (18) Kodera, T.; Yamazaki, T.; Masuda, M.; Ohnishi, R. Electrochim. Acta 1988, 33 (4), 537-540.
0 1990 American Chemical Society
Reduction of H 2 0 2on a Pt Electrode
The Journal of Physical Chemistry, Vol. 94, No. 16, 1990 6507
earlier in the studies on electrodissolution of metals. In this paper we show some types of oscillations which were observed during the reduction of hydrogen peroxide on a platinum electrode under galvanostatic conditions. The experiments were done in an aqueous solution of hydrogen peroxide and sulfuric acid using a rotating disk electrode. Honda et al?4 have previously studied oscillations during hydrogen peroxide reduction: they used a silver electrode and showed how the dissolution of silver contributes to the oscillations. T r i b ~ t s c h ~ and ' . ~ ~Tributsch and Bennett2' have investigated oscillations during electrocatalytic reduction of hydrogen peroxide on copper-ironsulfide electrodes. The reduction of hydrogen peroxide is, with the exception of the oxidation of hydrogen, perhaps one of the simplest electrocatalytic reactions. We show in this paper that oscillations can occur with this reaction even on a noble electrode. In contrast to the work of Honda et al., the electrode apparently acts only as a catalytic surface so that dissolution of the metal does not appear to play a role in producing the oscillations. Furthermore, we show that complex, yet still apparently regular, oscillations can arise, and we show the nature of these oscillations with the aid of attractor reconstruction. Exwriments The experiments were carried out with a platinum rotating disk electrode of radius 0.382 cm and associated rotator (Pine Instrument Co.). The counter electrode was a platinum foil of area 12.9 cm2. The reference electrode was mercurous sulfate (Radiometer America, Inc.). The three electrodes were placed in a three-neck 400" flask containing the solution. A Princeton Applied Research potentiostat/galvanostat Model 273 was used to control the current. The concentrations of hydrogen peroxide and sulfuric acid were 0.02-0.23 and 0.054.60 M, respectively. The rotation rate was held constant, usually at 1000 rpm but sometimes as low as 400 rpm or as high as 1600 rpm. In a few experiments noted below, the effect of rotation rate was studied by changing the rotation rate in a stepwise manner during the course of an experiment. Experiments were done galvanostatically with time zero taken at that moment when the galvanostat was turned on. The potential between the working and reference electrodes was monitored continuously on a recorder as well as digitally at lo00 Hz by means of a laboratory computer. Reactions The main cathodic reaction occurring on the rotating disk electrode is the reduction of hydrogen peroxide
H 2 0 2 + 2H+
+ 2e-
-
2H20
(1)
which has a standard potential of 1.77 V (NHE). The reversible potential is somewhat below this value for the lower H+ concentrations of our experiments; for example, at pH = 1.0 and [H202]= 0.05 M the reversible potential is 1.68 V (NHE). At the same concentration of sulfuric acid the reversible potential for the reduction of H+ 2H+
+ 2e-
-
H,
(2)
is -0.06 V (NHE). In the experiments described in this note, the 119) Yamazaki. T.: Kodera. T. Electrochim. Acta 1989. 34 (7). 969-975. (20) Shell, M.;Albahadily; F. N.; Safar, J.; Xu, Y. J . Phys.'Chem. 1989, 93, 4806-48 IO. (21) Buck, R. P.; Griffith, L. R. J . Electrochem. SOC.1962, 109 ( I l ) , 1 105-1 013. (22) Horanyi, G.; Inzelt, G.;Szetey, E. J . Electroanal. Chem. 1977. 81, 395-401. (231 Horanyi, G.; Rizmayer, E. M . J . Electroanal. Chem. 1983, 143, 323-336. (24) Honda, M.; Kcdera, T.; Kita, H . Electrochim. Acta 1986, 31 (3), 377-383. (25) Tributsch, H. Ber. Bunsen-Ges. Phys. Chem. 1975, 79 (7). 570-579. (26) Tributsch, H. Ber. Bunsen-Ges. Phys. Chem. 1975, 79 (7). 580-587. (27) Tributsch, H.; Bennett, J. C. Ber. Bunsen-Ges. Phys. Chem. 1976,40 (4), 321-327.
potential always oscillated between values such that reaction 1 should occur but reaction 2 should not. Anodic reactions on the platinum counter electrode are 2H20
-
H202
0, + 4H+ + 4e-
(3)
0,+ 2H+ + 2e-
which have standard potentials of 1.23 V (NHE) and 0.69 V (NHE), respectively. Both can occur under the conditions of our experiments. Results Potential oscillations occurred over ranges of concentrations, disk rotational rates, and applied current. We limited our study to concentrations of hydrogen peroxide and sulfuric acid of 0.02-0.23 and 0.05-0.60 M, respectively, to rotation rates of 400-1600 rpm and currents of 10-35 mA (current densities from 2 1.8 to 76.4 mA/cmZ). In all our experiments with this system there was, after some initial transient, a very slow change in oscillatory behavior with time. These changes occurred sufficiently slowly, however, such that the oscillations could be taken to be stationary and analyses, such as the application of Fourier transform methods or the reconstruction of attractors, could be carried out. As an example, consider the behavior shown in Figure I . A series of oscillation types is shown. After an initial transient small amplitude, relatively high frequency harmonic oscillations set in as shown in Figure la. Note that I / , s of the time series is shown, beginning a t a time 140 s after the beginning of the experiment. (Time zero is that moment when the galvanostat is turned on.) These oscillations slowly grow in amplitude, decrease in frequency, and become less harmonic, more relaxation-like as can be seen from Figure 1, b and c, for t = 170 s and t = 195 s, respectively. The single-peak oscillations of Figure la-c then give way to multipack oscillations as shown in Figure Id-f. The transition to multipeak oscillations can be seen more clearly with reference to Figure 2. Here the attractors (from the stationary segments of Figure 1) are shown in three-dimensional state space. They were constructed by using time delays of 2 X and 4 X s, r e s p e c t i ~ e l y . ~Figure ~ ~ ~ ~2a corresponds to Figure I C and thus shows in state space the limit cycle attractor for the one-peak oscillation which occurs around t = 195 s. The trajectories flow clockwise as viewed in Figure 2. Figure 2b shows the limit cycle at a somewhat later time. The cycle of Figure 2b has a pronounced peak in the lower right, and the flow is slower in that portion of the cycle. By Figure 2c an extra loop develops in the limit cycle. (Figure 2c corresponds to Figure Id, t = 265 s.) By Figure 2d ( t = 327 s corresponding to Figure le) several loops are evident in the cycle. There is apparently a saddle focus having one negative real eigenvalue and a set of complex eigenvalues with positive real parts which influences the flow on the attractor. The trajectories move slowly clockwise around the large outer loop, are injected toward the saddle focus, and then spiral outward before making another transit around the large outer loop. The change in behavior, then, as seen in Figure 2a-d (or Figure lc-e) is apparently caused by the approach of a saddle focus to the neighborhood of the limit cycle. Finally, an even more complicated behavior arises as shown in Figure I f for t = 690 s. There are now two types of largeamplitude peaks, those with sharp top and bottom and those with an extended lower portion. These two types of peaks alternate. All the behavior shown above, and throughout this paper, appears to be regular; Le., it appears to be periodic (although complex) with some modest variations because of noise. It is likely, however, that further study would yield low-order chaotic behavior. Other similar types of complex oscillations, obtained at somewhat different conditions, are shown in Figure 3. Figure (28) Packard, N . H.; Crutchfield, J. P.; Farmer, J. D.; Shaw, R. S. Phys. Reu. Lett. 1980, 45, 7 12-7 16. (29) Takens, F. In Dynamical Systems and Turbulence; Rand, D. A., Young, L. S., Eds; Lecture Notes in Mathematics 898, Springer: Heidelberg, 1981; pp 366-381.
Fetner and Hudson
6508 The Journal of Physical Chemistry, Vol. 94, No. 16, 1990 (e)
1
740
..
' 1700
14612
740
14037
1405
Time (Seconds)
(b)
"1 Y
240 1700
I 17012
17037
1705
Time (Seconds)
Figure 2. Attractors shown in three-dimensional phase space constructed using delays of 2 r and 4T where T = 1O-) s: (a) corresponds to Figure I C ( t = 195 s), (b) t = 235 s, (c) corresponds to Figure Id (t = 265 s), (d) corresponds to Figure l e (t = 327 s).
S I W
240
1950
19512
19537
1955
" 1
Time (Seconds)
(d)
-160
I
4
00
25
Time (Seconds)
75
1
10.0
(b) 640 7
"
I
I 2650
265'12
26537
W
I 2
240
2655
Time (Seconds)
w v
(e)
25
-160 0 0
75
100
Time (Seconds) Figure 3. Complex oscillations: (a) 0.03 M H202,0.10 M H2S04, 10 mA, 1000 rpm; (b) 0.09 M H202,0.27 M H2S04,30 mA, 1000 rpm.
I ' W
240 3270
327 12
32737
1
(f)
-
i
I
240
W
E
L Z
vi
640
3275
Time (Seconds)
Y
490
W
-160
4 20 0
240 6900
69075
69225
6930
Time (Seconds)
Figure 1. Time series of potential oscillations (0.09 M H202,0.19 M H2S04,I = IO mA, Q = 1000 rpm): (a) harmonic oscillations, t = 140 s, (b) t = 170 s, (c) t = 195 s, (d) I = 265 s, (e) t = 327 s, (f) t = 690
25 0
35 0
40 0
Time (Seconds) Figure 4. Single-peak oscillations (0.085 M H202,0.33 M H2S04,35 mA, 1000 rpm).
S.
3a shows oscillations with a single type of large-amplitude oscillation, that with the extended lower portion. In Figure 3b a more complicated waveform is shown. Single-peak, slow oscillations occurred under some conditions as shown in Figure 4. Note that the period is over 10 s, and this period increased slowly over the course of the experiment. An oscillation consisting of several small peaks and a single large peak is shown in Figure 5. In Figure 5 the sequence is four
8 25
Time (Seconds) 8 5 1 Figure 5. Time series (0.05 M H202, 0.20 M H2S04.10 mA, loo0 rpm).
J . Phys. Chem. 1990, 94, 6509-651 1 "
."
( a ) 1000 rDm
,
W
r 2 >
340
Y
W
640
00
I75
.
(b) 400 rpm
Time (Seconds)
5 25
70 i
h
W
r >
340
Y
W
50 0
51 75
5 5 25
5710
Time (Seconds) Figure 6. Effect of rotation rate (0.05 M H202. 0.10 M H2S04, 10 mA): (a) 1000 rpm, (b) 400 rpm.
small and one large. Note that the small peaks occur at low values of the potential, in contrast to those shown in Figure 1. The oscillations of the type shown in Figure 5 have been seen by many investigators in several types of systems and thus will not be discussed further here. See, for example, the recent extensive study of Schell et al. and the references therein.*O
6509
The effect of rotation rate is shown in Figure 6. The top portion (a) shows the potential measured with fl = lo00 rpm. The rotation rate was then decreased to 400 rpm, resulting in the behavior shown in (b). (There is a short transient between parts a and b of Figures 6.) The period has obviously increased although the general shape is unaltered. An increase in rotation rate to 1000 rpm returns the behavior to the oscillations shown in Figure 6a. In general, over the conditions of these experiments the frequency of oscillations increased with increasing rotation rate. This indicates that mass transfer is one of the resistances that control the rate of the electrocatalytic reaction under the conditions of the experiments described in this note.
Concluding Remarks We have shown that oscillatory behavior can occur for the reduction of hydrogen peroxide under galvanostatic conditions on a platinum electrode. Further investigation of this oscillating system appears to be warranted since it involves a relatively simple electrocatalytic reaction which yields interesting dynamic behavior. It is also likely, judging from the complexity of the oscillations reported in this paper, that low-order chaos can be found for some parameter values. Also note that a rich variety of oscillatory behavior has now been found in several types of electrochemical processes; electrochemistry is a fertile field for the applications of theories of nonlinear dynamics. Acknowledgment. This work was supported in part by grants from the National Science Foundation (CBT-8713070) and the Center for Innovative Technology, Commonwealth of Virginia. Registry No. H202,7722-84- 1; H2S04,7664-93-9; platinum, 744006-4.
COMMENTS Response to "Ludwig Boltzmann and the Norbornyi Cation" Sir: Recently, Kramer et al.' have made the novel proposal that molecules, such as the norbornyl cation, which undergo very rapid rearrangements returning to the same chemical structure so as to scramble their atoms (degenerate rearrangement processes), have much greater entropy and therefore more favorable free energy than similar molecules which are not rapidly scrambling. They contend that a molecule with a group of N interchanging atoms would be favored at equilibrium by a factor of N! over a similar but nonscrambling molecule since this is the number of permutations which would be distinct if the atoms could be individually labelled. The purpose of this paper is to show that the above idea is incorrect. The following statements by well-known theoreticians are relevant: "a permutation of indistinguishable objects cannot be regarded as a permutation at all."2a "Proceeding now to consider an assembly of N non-localized identical systems, we may suppose first that these are hypothetically labelled, 1, 2, ..., N . Then any particular microscopic state of the assembly is just one ( 1 ) Kramer, G. M.; Scouten, C. G.; Kastrup, R. V.; Ernst, E. R.; Pictroski, C. F. Ludwig Boltzmann and the Norbornyl Cation. J . Phys. Chem. 1989, 93. 6257. (2) (a) Golden, S. In Introduction to Theoretical Physical Chemistry; Addison-Wesley: Reading, MA, 1961; Chapter on the Maxwell-Boltzmann Method, p 99. (b) Rushbrooke, G. S. In Statistical Mechanics; Oxford University Press: London, 1949; p 38. (c) Gibbs, J. W. In Elementary Principles in Starisrical Mechanics; Yale University Press: New Haven, CT, 1902; Chapter XV, p 187.
0022-3654190 /2094-6509$02.50/0
member of a set of N! such states obtained from this one by permuting the labels 1, 2, ..., N: and these N ! states of the assembly are mutually distinguishable providing that labels are attached. Thus all the distinguishable microscopic states of the hypothetical assembly of labelled systems fall into sets of N! states, the members of any set being mutually distinguishable only on account of the labels artifically attached to the systems. Consequently, if we enumerate the complexions of N identical nonlocalized systems by the expedient of attaching hypothetical labels to the systems, then we count all the truly distinguishable states of the assembly N! times."2b "The essence of statistical equilibrium is the permanence of the number of systems which fall within any given limits with respect to phase. We have therefore to define how the term "phase" is to be understood in such cases. If two phases differ only in that certain entirely similar particles have changed places with one another, are they to be regarded as identical or different phases? If the particles are to be regarded as indistinguishable, it seems in accordance with the spirit of the statistical method to regard the phases as identical."2c The above opinions seem conclusive; however, resort to authority is not the most satisfactory form of scientific argument. It would be better to construct a proof from a commonly agreed on set of first principles; but, the Kramer paper effectively does not accept the above statements which might well be considered as stemming from a fundamental postulate by others. In other words, these authors seem to be starting from different basic ideas then have others before them. Although the authors do not state this directly, the entropies that they predict for various symmetrical and unsymmetrical systems seem to indicate that they do not regard 0 1990 American Chemical Society
