2356
J. Phys. Chem. 1991, 95, 2356-2361
recently been identified in the BR photocycle, K’ having been proposed in the quantitative analysis of picosecond, timeresolved absorption and fluorescence ~igna1s.I~Both species have similar vibrational relaxation rates ( a 7 ps for BR’ and -8 ps for K’I3). Efforts to extend the anti-Stokes R R spectroscopy described here to directly monitor K’ are underway.
Acknowledgment. This research is supported under a grant from the National Institutes of Health (GM36628). G.H.A. expresses his appreciation for support as a Senior Alexander von Humboldt Awardee and for the hospitality of Professor E. Schlag at the Technical University of Munich where part of this paper was prepared.
Coexlstlng Cyclic Voltammograms Cyan Ranjan Parida and Mark Schell* Department of Chemistry and Center for Nonequilibrium Structures, Southern Methodist University, Dallas, Texas 75275 (Received: March 27, 1990; In Final Form: October 3, 1990)
Cyclic voltammetry was used to study the electrochemicaloxidation of formate/formic acid at a rotating platinum disk electrode. The electrochemical system, subjected to a cycling potential, is interpreted as a periodically forced nonlinear system, and cyclic voltammograms, which repeat, are interpreted as projections of limit cycles. Measurements show that, for a range of values of the lower potential limit, two limit cycles coexist. A reaction of the oxidation process in which strongly bound intermediates and chemisorbed hydroxyl radicals are reactants is postulated as being strongly nonlinear, and this postulate is used to explain the results. General implications of the results are also briefly discussed.
Introduction Chemical instabilities, such as the onset of oscillations or the occurrence of multiple stationary states, take place in a variety of electrochemical processes. A large number of examples exhibiting these types of electrochemical instabilities are discussed in reviews that cover the older literature.’.* Recently, more complex electrochemical instabilities, which include transitions into chaos3*‘and hyperchaos,5 the onset of quasiperiodic oscillations? and bifurcation sequences of mixed-mode oscillatory states,’Vs have been discovered. Instabilities arise in processes that are nonlinear and that take place in systems held far from equilibrium.2 There are several possible sources of nonlinearities in electrochemical processes: rate coefficients can possess a nonlinear dependence on the potential difference between an electrode and the surrounding solution; rate laws can be nonlinear in electrode-surface concentrations; nonlinearities can be present in fluxes associated with transport across the boundary layer that separates an electrode surface from the bulk solution. In this paper, we use the results of experimental measurements to illustrate that underlying nonlinearities can drastically affect the results of a standard electrochemical technique, cyclic voltammetrye9 Cyclic voltammetry is a tool now commonly employed in several fields. In particular, its usage is widespread in physical chemistry. Examples of application of cyclic voltammetry recently published in this journal include the characterization and analysis of lipid membrane-semiconductor junctions,1° electrooxidation pathway
of simple organic molecules,” chemisorbed C O on Pt(l1 1 ) , 1 2 supercritical fluids,” ion transfer across liquid-liquid phase boundaries,I4 and adsorption and reactions of disulfonated anthraquinones at mercury electrode^.'^ In cyclic voltammetry, the potential (difference) between a working electrode and a reference electrode is changed so that a range of values between two reversal points is repeatedly traversed on a linear time scale. The current is measured during the experiment. The curve generated in the current-potential plane during one complete cycle over the specified range of potential values is called a cyclic voltammogram. Cyclic voltammograms that repeat are usually obtained after a finite number of traversals and contain useful information concerning processes at the electrode. The position, shape, and height of current peaks, as well as the number of such peaks, can often be used to derive approximate values of rate coefficients, identify kinetic steps of an electrochemical reaction, and identify chemical species that appear through either purely chemical or electrochemical reactions. A cyclic voltammogram that repeats and that is stable with respect to small perturbations is equivalent to a limit cycle that is projected from state space onto the current-potential plane.a’6J7 Furthermore, a nonlinear electrochemical process, subjected to a cyclic sweeping potential, is a periodically forced nonlinear (10) Zhao, X. K.; Subhash, B.; Fendler, J. H. J . Phys. Chem. 1990,94, 2043. (11) Leung, L.-W. H.; Weaver, M. J. J . Phys. Chem. 1989, 93, 7218. (12) Zurawski, D.; Wasberg, M.; Wieckowski, A. J. Phys. Chem. 1990, 94. 2076. (13) Niehaus, D.; Philip, M.; Michael, A.; Wightman, R. M. J . Phys. Chem. 1989, 93,6232. (14) Wandlowski, T.; Marecek, V.; Holub, K.;Samec, Z . J . Phys. Chem. 19R9. .-. , 93. - - , R2M - - - .. (15) He, P.; Crooks, M. R.; Faulkner, L. R. J . Phys. Chem. 1990, 91, 1135. (16) Nonequilibrium Dynamics in Chemical Systems; Vidal, C., Pacault, A,, Eds.; Springer: New York, 1985. (17) State space (or phase space) is a space in which the coordinates are all the possible observables that can evolve in time. In the system considered here, the observables include the concentration of species on the surface of the electrode, which is why we refer to the ‘projection into the current-potential plane”. In state space, a trajectory (or orbit) represents the evolution of the system and, since we are dealing with a dissipative system, almost all trajectories approach an attractor. A limit cycle, which is the representation in state space of an asymptotic periodic solution, is an example of an attractor. Detailed discussions on this subject can be found in refs 2 and 16. I~
(1) Hedges, E. S.;Myers, J. E. The Problem of Physical Periodicity; Arnold: London, 1926. (2) Nicolis, G.; Portnow, J. Chem. Reo. 1973, 73, 365. Wojtowicz, J. In Modern Aspects of Electrochemistry; O M Bockris, J., Conway, B., Eds.; Plenum: New York, 1973; Vol. 8, p 47. (3) Bassett, M. R.; Hudson, J. L. J . Phys. Chem. 1988, 92,6963. (4) Schell, M.; Albahadily, F. N. J. Chem. Phys. 1989, 90,822. (5) Diem, C. B.; Hudson, J. L. AIChE J. 1987, 33, 218. (6) Basset, M.R.; Hudson, J. L. Physica D 1989, 35, 89. (7) Jaeger, N. 1.; Plath, P. J.; Quyen, N. Q.In Temporal Order, Rensing, L., Jaeger, N. I., Eds.; Springer: Heidelberg, Germany, 1985; pp 103-104. (8) Schell. M.;Albahadily, F. N.; Safar. J.; Xu,Y. J. Phys. Chem. 1989, 93,4806. (9) Bard, A. J.; Faulkner, L. R. Electrochemical Methods Fundamentals and Applications; Wiley: New York, 1980.
0022-3654/91/2095-2356$02.50/0
0 1991 American Chemical Society
Coexisting Cyclic Voltammograms
The Journal of Physical Chemistry, Vol. 95, No. 6,1991 2351
system.18 More precisely, due to potential-dependent rate coefficients, it is a parametrically forced system.Ig Although the potential, as a function of time, is a function that might be considered more complicated than the usual sinusoidal term, it is nevertheless a periodic forcing term (a triangular function). Establishing the relationship between cyclic voltammetry and forced nonlinear systems begs an obvious question: Can the same types of nonlinear dynamical phenomena observed in other forced nonlinear systemst8be obtained by using cyclic voltammetry? We use the results of an experimental study to demonstrate that one such phenomenon does occur: the coexistence of two limit cycles.'8 The process we have studied using cyclic voltammetry is the oxidation of formate/formic acid at a platinum e l e ~ t r o d e . ~ J ~ * * ~ ~ A region of bistability is characterized in which two different families of cyclic voltammograms coexist.
Experimental Section A Pine Instrument Model RDE4 potentiostat was used to control the potential and measure the current. Measurements were recorded with an x-y recorder (Houston Instrument 2000 recorder). In some of the experiments, measurements were also directed to a Hewlett-Packard (HP) Model-9237 computer through an H P Model-3852A data-acquisition unit equipped with a HP Model-44702A 13-bit voltmeter. The electrochemical cell was a SOO-mL, three-neck flask. The flask was thermostated at 25.0 f 0.2 "C and contained a 400-mL solution consisting of 1.0 M sodium formate (Fisher Scientific) and lo4 M sulfuric acid. Millipore-processed distilled water was used in all solutions. A saturated calomel electrode (Beckman) and a platinum wire served as the reference electrode and counter electrode, respectively. All potentials are reported with respect to the reference electrode. A rotating disk, 18.0 mm in diameter, 8.0 mm polycrystalline platinum, 10.0 mm Teflon (Pine Instrument Model No. AFDI 1580, Serial No. 461 9), was employed as the working electrode. The disk was attached to a Pine Instrument ASR-type rotator and rotated at 3000 rpm. The working electrode was cleaned electrochemically in a solution of 1.0 M H 2 S 0 4 before each set of experiments. The cleaning procedure consisted of passing the electrode through the following steps: the potential was held at a value of 2.0 V for 40 s then at 0.0 V for 5 s, at -0.4 V for 10 s, and then finally at 0.0 V for 5 s. This procedure was repeated 30 times. Results and Discussion Large- and Small-Amplitude Cyclic Voltammograms. Several cyclic voltammograms, measured at consecutively higher values of the lower potential limit (Ipl), are shown in Figure 1. Each voltammogram repeated, i.e., a sufficient number of traversals were completed before recording so that transient behavior was not present. A different set of cyclic voltammograms, which were also measured after transient behavior disappeared, is shown in Figure 2. For convenience, we call repeating cyclic voltammograms of the type shown in Figure 1 large-amplitude CVs and of the type in Figure 2 small-amplitude CVs. None of the cyclic voltammograms in Figures 1 and 2 is that unusual. What might be regarded as unusual is the fact that the cyclic voltammogram in Figure 1b and the cyclic voltammogram of Figure 2c were both ~
(18) Stoker, J. J. Nonlinear Vibrations; Interscience: New York, 1950.
Hayashi, C. Nonlinear Oscillations in Physical Systems; McGraw-Hill: New York, 1964. Holmes, P. J.; Rand, D. A. J . Sound Vibr. 1976, 44, 237. Huberman, B. A.; Crutchfield, J. P.; Packard, N. H. Appl. Phys. Lett. 1980, 37. 750. Kapral. R.; Schell, M.; Fraser, S . J . Phys. Chem. 1982, 86, 2205. ( 1 9 ) McLachlan, N. W . Ordinary Non-linear Duferential Equations in Engineering and Physical Sciences; University Press: Oxford, 1958; Chapter VII. Levan, R. W.;Koch, B. P. Phys. Lett. A 1981, 86, 71. (20) Buck, R. P.; Griffith, L. R. J . Electrochem. Soc. 1962, 109, 1005. (21) Rhodes, D. R.; Steigelmann, E. F. J . Electrochem. Soc. 1965, II2, 16.
(22) Sun, S.G.; Clavilier. J.; Bewick, A. J. ElectmnuI. Chem. 1988, 240, 147. (23) Corrigan, D. S.; Weaver, M. J. J. Elecrrwnul. Chem. 1988,241, 143. (24) Kita, H.; Katagiri, T.; Kunimatsu, K. J. E~~Ctfocmcl~. Chem. 1987, 220, 125. (25) Kunimatsu, K.; Kita, H. J . Electroanal. Chem. 1987, 218, 155.
I
I
I 500 P O T E N T I A L (mV)
I 1000
I
I 1000
h
4
E
v
3
u
500
P O T E N T I A L (mV)
-
P O T E N T I A L (mV) Figure 1. Set of cyclic voltammograms recorded at different values of the lower potential limit (Ipl). Current is plotted against potential (SCE). Sweep rate = 50.0 mV/s, upper potential limit = 1OOO.O mV (SCE). (a) Ipl = 20.0 mV. At this value for the Ipl, the system relaxes to the large-amplitude CV from all starting points. (b) Ipl = 160.0 mV. (c) lpl = 210.0 mV. The latter cyclic voltammogram exists in the vicinity of a critical value for the Ipl. A drastic change of behavior occurs at that critical value. A cyclic voltammogram of the type shown either does not exist or is not stable at values of the Ipl that are greater than this critical value.
measured under the exact same experimental conditions. This result is evidence that large-amplitude CVs and small-amplitude CVs can coexist. The measurements, Figures 1 and 2, interpreted within the context of the theory of suggest that subjecting the electrochemical system to a cycling potential can lead to a bistable system. We now describe the implementation of an experiment that led to results that are consistent with predictions of bifurcation theory regarding a bistable system. The exprimental method can be applied to any electrochemical process and hence
2358 The Journal of Physical Chemistry, Vol. 95, No. 6, 1991
Parida and Schell
1
4
0.lC 1 0
POTENTIAL (mV)
1 POTENTIAL (mV)
0. h
U E
w
Ea
0.
a 3 u 0.
POTENTIAL (mV) Figure 2. Sequence of small-amplitude cyclic voltammograms shown at successively smaller values of the Ipl. All other conditions the same as in Figure 1. (a) Ipl = 260 mV. Shown on the same scale as used in Figure 1. (b) The same as (a), but on a largr scale. (c) Ipl = 160 mV.
This cyclic voltammogram was found to exist under the same conditions as shown in Figure 1b. Initial conditions determine which one the system relaxes toward during the sweeping process. (d) Ipl = 110.0 mV. can be used to determine whether the process, subjected to a cycling potential, is bistable. After cleaning the working electrode, the potential was cycled between 0.0 and 1000 mV at a rate of 50 mV/s. After approximately 50 cycles, a large-amplitude CV repeated. The procedure that was followed next consisted of increasing the value of the lpl in small increments (120mV). At each new value of the Ipl, the potential range was traversed until a voltammogram was obtained that repeated. Only large-amplitude CVs were obtained until the lpl was changed to the value of 220.0 mV. At this latter value of the lpl the system exhibited a drastic change. The amplitude of the cyclic sweeps decreased until a small-amplitude CV was obtained. The lpl was then increased several times and then decreased. The maximal magnitude of the decrements applied to the Ipl was 10.0 mV. Small-amplitude CVs were obtained at each new value of the Ipl until it was decreased to 70.0 mV. At this value of the Ipl, several cyclic voltammograms were obtained
POTENTIAL LIMIT (mV) Figure 3. Measured peak currents of cyclic voltammograms that repeat. The points on the upper curve correspond to large-amplitude CVs; see Figure 1. The system was moved along the upper branch by increasing the Ipl. After moving to the point D, the system relaxed to a small-amplitude CV of the type shown in Figure 2. The system was moved along the lower branch by decreasing the Ipl. Each point on this lower branch corresponds to a small-amplitudeCV. When the system was moved to the point U, it appeared that for several cycles the system would relax to a small-amplitude CV, but instead, it eventually relaxed to a largeamplitude CV. with small amplitudes but it was evident that the amplitude was slowly increasing during the cycling process. Eventually, large increases in the amplitude were observed, and then, finally, the system relaxed to a large-amplitude CV. The results of the experiment are shown in Figure 3, where peak amplitudes of the cyclic voltammograms are plotted against the value of the lpl. The upper branch of points corresponds to large-amplitude CVs and the lower branch to small-amplitude CVs. There is a substantial range of values for the Ipl in which part of the upper branch and part of the lower branch coexist. The experiment was repeated several times, and a region of coexistence was always obtained. The result does not appear to be a simple extrapolation of bistability in the underlying constant potential system. The constant-potential system (zero sweep rate) exhibits a monotonic curve consisting of stationary states. A region of coexistence was obtained only at finite sweep rates. Results of Perturbing the Cyclic Voltammograms. The fact that the response, after each change in the lpl, relaxed to a cyclic voltammogram that repeated is consistent with the idea that such a cyclic voltammogram corresponds to a projection of an attractor, i.e., an attracting limit cycle. When more than one attractor exists in state space, boundaries divide state space into basins. A one-to-one correspondence exists between basins and attractors, and each basin includes the complete neighborhood of one, and only one, attractor. All initial conditions that correspond to points in a basin of an attractor give rise to temporal evolutions that (ignoring the effects of fluctuations) approach the asymptotic state that is associated with that attractor. It follows that if the system is perturbed away from an attractor but remains in that attractor's basin, the system will follow a path that returns it to the neighborhood of that attractor. However, if the system is perturbed away from an attractor in such a way that it is moved across a boundary into a basin of another attractor, the system will then follow a path that approaches this latter attractor. The applicability of these ideas from dynamics theory to the system under consideration was tested. The system was allowed to relax until a cyclic voltammogram was obtained that repeated. Under conditions in which both a large-amplitude CV and a small-amplitude CV were obtained, the application of perturbations could cause the system to change from one type of cyclic voltammogram to the other. The results of such experiments are represented in Figures 4 and 5. Current is plotted against time in Figure 4. The large-amplitude oscillations in the first part of the time series correspond to cycles in the current-potential plane that traverse a large-amplitude CV. A perturbation was applied by interrupting the sweeping process and holding the value that the potential possessed (-390 mV) at the time of the interruption constant for a period of approximately 300 s. The perturbation
The Journal of Physical Chemistry, Vol. 95, No. 6, 1991 2359
Coexisting Cyclic Voltammograms
I
I
I.
II h
h
a
4
w
w
E
E
;1
ez; ez;
3
3
u
U
-5 I
TIME (s) Figure 4. Perturbation experiments. Current is plotted against time. The large-amplitude oscillations are observed in this temporal representation when a large-amplitude CV is obtained in the current-potential plane. The first break in the sequence of large-amplitude oscillations is a result of holding the potential constant for approximately 300 s. The perturbation did not move the system out of the basin of the attractor associated with the large-amplitude CV. The second break in the sequence of large-amplitude oscillations is a result of holding the potential constant for approximately 1020 s. The system then relaxed to a state in which it exhibited small-amplitude oscillations. Ipl = 150 mV. I
b
I I J 500 1000 POTENTIAL (mV) Figure 5. Induced transition between cyclic voltammograms. The system was prepared so that it was brought to a state in which it exhibited a small-amplitude CV, labeled 1. After applying a current pulse, the system relaxed to the state in which it exhibited a large-amplitude CV, labeled 2. Only the first few transient cycles are shown. Ipl = 130 mV.
was initiated as the current was passing through the vicinity of its maximum in the forward sweep. Following the resumption of the sweeping process the system was observed to relax back to the large-amplitude CV or, in the representation of Figure 4, the large-amplitude oscillations. The system was again perturbed, but this time the potential was held constant for greater than loo0 s. After the sweeping process was resumed, the system relaxed to a small-amplitude CV; see Figure 4. The results of a different perturbation experiment are represented in Figure 5 . In this case the system was prepared so that it exhibited a small-amplitude CV, which is labeled in Figure 5 with the number 1. The sweeping process was interrupted by applying an anodic current spike with a magnitude of 30 mA. After the perturbation was applied, the system relaxed to a large-amplitude CV, which is labeled with the number 2. Relation to Mechanism. Next, we describe the cyclic voltammograms and then present a possible explanation for the bistability. First consider the large-amplitude CVs, Figure 1. Beginning at the low-potential limit, the CVs exhibit an ascending anodic curve. The oxidation of formic acid/formate is responsible for this part of the anodic current.20 At larger values of the potential, the curve passes through a maximum value and then descends to considerably smaller values of the current. In the region of small-current values, oxides form on the working electrode surface.20 Although the rate of the oxidation of the
I1
I 1 500 100 POTENTIAL (mV) Figure 6. Large-amplitude CV at a lower rotation rate (300 rpm). Ipl = 0.0 mV. A shoulder appears in the cyclic voltammogram at lower rotation rates.
formic acid/formate on platinum oxide is not comparable to that on a bare platinum electrode, it does occur.20,21On the reverse sweep the oxidation of formate/formic acid and the reduction of platinum oxide combine to yield a net anodic current. An extremely sharp rise in current occurs on the reverse sweep at a value of the potential close to the value where the last of the oxide layer is removed.20 The cyclic voltammogram in Figure I C was the last measured large-amplitude CV recorded before the transition to the lower branch, Figure 3. When the lpl is increased beyond a critical value, oxides are maintained on the electrode surface and the large-amplitude CV disappears. Note that, for the case of the CV in Figure IC, the potential is moving in the forward direction before the sudden increase in current occurs, which indicates that the lpl is within a response time of the potential at which oxides are removed. Details contained in previously published voltammograms are not seen in the large-amplitude CVs. The current density is larger than that obtained with a stationary electrode due to the high rotation rate. The high rotation rate causes a larger rate of transport of fuel to the electrode surface, and consequently the peak in the forward scan hides some details. At lower rotation rates, or with stationary electrodes, a shoulder is observed;20see Figure 6 . Oxide layers are maintained throughout the cyclic sweep in the case of the small-amplitude CVs. From comparison of the small-amplitude CVs, Figure 2, it is evident that at the low end of the potential range, higher currents are achieved with smaller values of the lpl. This is simply a result of moving farther into the potential range and consequently spending more time within this range, at which oxides are removed. On decreasing the lpl, a critical value is eventually reached at which all the oxide layers can be removed. To discuss the possible cause of the bistable behavior, we present a brief overview of the current understanding of the mechanism. We restrict the discussion to the oxidation of formic acid. There is some evidence that the oxidation of formate and the oxidation of formic acid exhibit similar behavior.” It is now widely accepted that both the oxidation of formic acid and formate ions follow at least two parallel paths. A mechanism for one of the paths in the oxidation of formic acid, which we call the direct path, consists of the following steps:22*2s*26
--
(1)
C02
(3)
HCOOH HCOOH(a) COOH
-
H
+
HCOOH(a) COOH + H
+ H+ + eH+ + e-
(2) (4)
(26) The mechanism is a combination of steps given in refs 22 and 25. Adsorption of a molecule without chemical bond formation is represented by labeling the adsorbate with an a; an atom in a molecule that is italicized indicates bond(s) formation between that atom and platinum.
2360 The Journal of Physical Chemistry, Vol. 95, No. 6, 1991 Several modifications of this proposed mechanism for the direct path, as well as additional steps, exist in the literat~re.~'-*~ In a second path, an intermediate is formed that, at sufficiently low potentials, remains essentially inert and thus blocks the direct path. A comparison between the voltammetric behavior of carbon monoxide and formic acid provided evidence quite some time ago that CO was an intermediate; see, for example, the discussion by Brummer and M a k r i d e ~ . ~However, ~ other experimental measurements suggested other possibilities for the strongly adsorbed i~~termediate.~'-~ Only through the recent application of surface spectroscopies has convincing evidence been obtained that overwhelmingly supports different forms of carbon monoxide as the identities of strongly adsorbed intermediates.' Several different kinetic steps have been proposed for the formation of C O that i n ~ l u d e ~ ~ , ~ ~ COOH H + CO + H2O (5) 1922*23*25*27
+
2COOH
- -+ CO
HzO
CO
HCOOH(a)
+ C02
(6)
+ H20
(7)
As with the direct path, the exact way C O is formed is not that
important to our arguments. However, we believe the process by which CO is removed from the electrode is directly related to the cause of the bistability. The efficient removal of CO requires the chemisorption of hydroxyl radicals; i.e.
- + + + - + +
H20
OH
H+
e-
(8)
The hydroxyl radicals in turn react with carbon monoxide: OH
CO
H+
C02
e-
(9) It is important to note that reaction 9 is widely used to represent the first step in the initial stage of the process by which an oxide layer is formed. The process by which an oxide layer is formed is complex and proceeds in several stages, each with steps in which many of the details remain unknown.30 We represent the steps that follow reaction 8 by the following chemical equations: OH 0 + H+ + e(10)
-
0-
(11)
Although the occurrence of reaction 10 is widely accepted, the pair of reactions 10 and 11 should be thought of as only a schematic representation of the steps in the process by which an oxide layer is formed. The kinetics, as formulated above, lend themselves to a simple explanation of the results. In the case of the large-amplitude CV, reactions of the direct path, eqs 1-4, cause the rise in the current during the first part of the forward sweep. During this part of the sweep C O is also formed, e&, eq 7, which inhibits the direct path. However, when a sufficiently large potential is reached, OH is chemisorbed, reaction 8, which in turn reacts with CO, reaction 9, and these two reactions contribute to the current as well as maintain vacant sites on the electrode that allow the continuation of the reactions in the direct path. If initial conditions are such that enough carbon monoxide is formed on the surface of the electrode under conditions in which bistability occurs, reaction 9 will dominate a portion of the forward sweep that is sufficiently large to cause a delay in the occurrence of the reactions represented by eqs 10 and 1 1. Consequently, the oxide formation only reaches a stage that can be removed on the reverse sweep. On the other hand, if initial conditions are such that the sweeping process begins at a sufficiently high potential and with a sufficiently high coverage of hydroxyl radicals on the surface of the electrode, reactions 9-1 1 will be the predominante reactions during the forward sweep. In the latter case, the oxide layer develops to a stage that cannot be completely removed on the reverse sweep. (27) Parsons, R.; VanderNoot, T. J . Electroanal. Chem. 1988, 257, 9. (28) Capon, A.; Parsons, R. J . Electroam/. Chcm. 1973, 45, 205. (29) Brummer,S. B.; Makridcs, A. C. J . Phys. Chem. 1964. 68, 1448. (30) Angerstein-Kozlowska, H.; Conway, 9. E.; Sharp, W. B. A. J . Elcctrwnal. Chem. 1973, 257, 9. Hoare, J. P. J . Electrochem. Soc. 1985, 132, 301. Conway, 9. E.; Liu. T.-C. Langmuir 1990. 6, 268.
Parida and Schell However, for practical reasons, the two types of behavior can only coexist for a finite range of values for the lpl. There must exist a critical value such that, for values of the lpl greater than this critical value, oxides cannot be removed regardless of the initial condition. There must also exist a critical value such that, for all values of the lpl less than this value, all oxides can be removed. The above explanation should be considered as oversimplified. A more detailed description of the mechanism should include the reactions of the formate ion and the reactions involving the fuel on the oxide layer. More importantly, in regards to the fundamental aspects of the explanation, a nonlinear feedback is necessary in order to obtain a bistable response.z18 It is true the rate of the reaction in eq 9 is proportional to both the concentration of CO and of OH. However, this is a relatively weak nonlinearity in the sense that the rate of reaction decreases as CO is removed. Since water molecules are more numerous than molecules of fuel, hydroxyl radicals, through reaction 8, will on the average replace the removed CO. Thus, without some type of feedback, one would expect the system to relax to a state where reactions 10 and 11 dominate the sweeping process. There does exist evidence that the reaction between O H and CO is more explosive, i.e., a reaction for which the rate can suddenly become large. Spectroscopic data show that most of the CO is suddenly removed when a potential is achieved at which hydroxyl radicals are chemisorbed;"*23 and also, an explosive reaction is consistent with data obtained in studies of oscillations in the potential under conditions of constant ~ u r r e n t . ~An ' assumption, made p r e v i ~ u s l y , ~that ' . ~ ~the rate of the reaction in eq 9 is proportional to the surface concentration of vacant sites provides such an explosive reaction. With this assumption, the rate law for reaction 9 takes the form rate = k(OH)(CO)S where round brackets denote surface concentrations, S represents the surface concentration of vacant sites, and k is a potentialdependent rate coefficient. Although this rate law is probably a crude approximation to the actual rate law, it is consistent with measured data from experiments that indicate an explosive rea ~ t i o n . ~If' the rate of reaction 9 is proportional to the density of vacant sites and since the reaction also produces a t least two vacant sites for every molecule of C 0 2 produced, the reaction is autocatalytic and thus contains a feedback mechanism. The more vacant sites produced, the faster the reaction. Consequently, such a reaction can remove OH faster than it is chemisorbed through reaction 8, even at low concentrations of CO. Therefore the reaction can prevent the system from relaxing to a state where reactions 10 and 11 dominate.
Summary and Some Speculative Remarks We have shown, through application of cyclic voltammetry on the oxidation of formate/formic acid at a rotating platinum disk electrode, that two different repeatable cyclic voltammograms can coexist under the same experimental conditions. The response of the system was consistent with the idea that the cyclic voltammograms are projections of attracting limit cycles. At this time it is difficult to ascertain how prevalent this phenomenon is in other electrochemical processes. We expect that different cyclic voltammograms can be found to coexist in some of the many other oxidation processes that are also coupled through reactions involving the same strongly bound intermediate (CO) to the process by which an oxide layer is formed.33 More general, but speculative, arguments exist for believing that bistability might be observed in other cyclic voltammetric experiments involving adsorbed intermediates. As stated, cyclic voltammograms are projections of limit cycles and, as noted, electrochemical processes are nonlinear. Coexisting limit cycles are commonly found in periodically forced nonlinear s y s t e m ~ . ' ~ J ~ (31) Xu, Y.; Schell, M. J . Phys. Chem. 1990, 94,7137. (32) Wojtowicz, J.; Marincic, N.; Conway, B. E. J . Chem. Phys. 1968,48, 4333. (33) Leung, L.-W. H.; Weaver, M. J. Langmuir 1990,6, 323.
J. Phys. Chem. 1991, 95, 2361-2364 Of course there is an obvious question: Why have coexisting cyclic voltammograms not yet been readily observed? The methods of finding and analyzing bistability in cyclic voltammetric experiments are not well-known. In this paper we have presented a method that is simply a generalization of methods applied to other systems. While the sweeping process is conducted an experimental constraint (here it was the lower potential limit) is
2361
varied over a wide range of values. Checking for bistability in cyclic voltammetric experiments is important. Uncovering bistability can lead to more information regarding the, electrode process such as the presence of nonlinear feedback mechanisms. Acknowledgment. This research was sponsored by the Robert A. Welch Foundation, Grant No. N-1096.
Absolute Number Density Calibration of the Absorption by Ground-State Lead Atoms of the 283.3-nm Resonance Line from a High-Intensity Lead Hollow Cathode Lamp and the Calculated Effect of Argon Pressures John W. Simons,* Roy E. McClean, Department of Chemistry, New Mexico State University, Las Cruces, New Mexico 88003
and Richard C. Oldenborg Chemical and Laser Sciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (Received: April 25, 1990; In Final Form: August 16, 1990)
The absolute number density calibration for the absorption by ground-state lead atoms of the 283.3-nm resonance line from a high-intensity lead hollow cathode lamp (Photron superlamp) is determined and found to be the same as that of a standard hollow cathode lamp. Comparisons of the calibrations to theoretical calculatioins are found to be quite satisfactory. The effects of argon pressures in the absorption cell on the calibration are examined theoretically by using a simple Lorentzian broadening and shifting model. These calculations show the expected reduction in sensitivity and increasing linearity of Beer-Lambert plots with increasing argon pressure.
Introduction We recently reported an accurate absolute number density calibration for absorption by ground-state lead atoms of the 283.3-nm line from a Westinghouse lead hollow cathode lamp.' The calibration was in excellent agreement with a theoretical model based on a Doppler-limited Fourier transform spectrum2 of the 283.3-nm line of the lamp. Since the lamp was to be used for some fairly rapid time-dependent lead atom number density measurements, as high a lamp intensity as possible was desirable. For this reason we decided to use a high-intensity hollow cathode lamp (Photron superlamp), for which an accurate calibration was required. A comparison of this high-intensity lamp calibration to our previous low-intensity lamp calibration is of interest. In addition, the most recent JANAF Tables contain revised lead atom vapor pressure value^.^ The effect of this revision on our earlier calibration and theoretical model is of interest. Since the experiments for which the high-intensity lamp will be utilized involve temporal lead atom number density determinations in a cell containing various pressures of argon, it is necessary to quantitatively correct the calibration, determined in the absence of argon, for pressure broadening and shifting of the absorption profile by argon. The method of determining this correction and its effect on the calibration are also presented here. It would be exceedingly difficult to experimentally calibrate the ~~~~
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(1) Simons, J. W.; Oldenborg, R. C.; Baughcum, S. L. J. Phys. Chem. 1989, 93. 1338. (2) Simons, J. W.; Palmer, B. A.; Hof,D. E.; Oldenborg. R. C. J. o p r . Soc. Am. B 1989,6, 1097. (3) Chase Jr.. M. W.; Davies, C. A,; Downey, Jr., J. R.; Frurip, D. J.; McDonald, R. A.; Syverud, A. N. JANAF Thermochemical Dara, 3rd ed.; J . Phys. Chem. Re/. Dara 1985, 14, Suppl. 1, 1756.
0022-3654/91/2095-2361.$02.50/0
absorption due to Pb(g) in the presence of inert gas since diffusional problems would not allow equilibrium vapor pressures of Pb(g) to be attained.
Experimental Section The high-intensity lead hollow cathode lamp was a Photron superlamp, serial no. 7668, operated at 7 mA with a boost current of 12 mA. The Westinghouse hollow cathode lamp was described earlier.' The absorption cell, apparatus, and procedure used in determining the calibration curve for the Photron superlamp are identical with that described previously for the Westinghouse lamp.' Results and Discussion Lamp Calibrationsfor Pb(g) Absorption without Added Argon. It is seen in Figure l a that the data for the Westinghouse lamp, presented earlier,' are shifted to slightly higher nl values when the JANAF Tables3 revised Pb( 1) vapor pressure data are utilized. The calibration data for the superlamp are presented in Figure 1b. Each point is the average of 4-6 repetitive measurements that were quite reproducible. These data were taken during alternate heating and cooling cycles of the Pb(1) sample. It is seen in Figure l b that the superlamp gives essentially the same calibration curve as the Westinghouse lamp. This agreement between the calibration curves for the two lamps indicates that their source functions are the same. We have shown previously'** via the Fourier transform spectrum of several lines2 and the 283.3-nm absorption calibration' for the Westinghouse lamp that the 283.3-nm source function is well represented by the expected set of five isotopic and hyperfine lines, the shapes of which are well represented by 750 K Doppler functions with little, if any, self@ 1991 American Chemical Society
