Energy Dynamics in the Bray−Liebhafsky Oscillatory Reaction - The

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Energy Dynamics in the Bray-Liebhafsky Oscillatory Reaction Dragomir R. Stanisavljev* Faculty of Physical Chemistry, Studentski trg 12-16, P.O. Box 47, 11158 Belgrade, Serbia ReceiVed: September 14, 2009; ReVised Manuscript ReceiVed: NoVember 2, 2009

Energy dynamics of the well-stirred, isothermally conducted Bray-Liebhafsky reaction is followed by monitoring the population of the first two vibration states of hydrogen peroxide. Excitations are detected by Raman spectroscopy showing periodical changes of the energy flow through the system, matching the periodicity of chemical oscillations. Well before chemical oscillation, rearrangement of energy provokes excessive excitation of the first vibration state of hydrogen peroxide followed by the phase-shifted excitation of the second state. The observed populations of excited states highly exceed equilibrium values, suggesting that the nonequilibrium distribution of energy related to the peculiar hydrogen bond network dynamics may be an important part of the reaction mechanism. Introduction Oscillatory chemical reactions belong to one of the most interesting phenomena because they demonstrate spontaneous appearance of order in the investigated system. As a result of the specific coupling with the surroundings, a periodic increase of the intermediate concentrations appears, with a relatively easily observed time ordering. Although spontaneous ordering has a well-established base in the thermodynamics of highly nonequilibrium, opened systems,1,2 detailed reaction mechanisms of these processes are still the subject of intensive research, especially in living organisms.3,4 The main problem in the experimental investigations of such peculiar systems is a lack of sufficiently fast and sensitive measuring techniques accompanied, usually, by low selectivity and not easily available experimental equipment. All these make difficulties in detecting reactive intermediates and defining elementary steps involved in the process. Because of this, the help of theory is needed. The toretical approach to the problem is aimed at finding a set of chemical equations able to simulate oscillatory evolution. As the kinetic descriptions of the proposed models are related to the systems of highly nonlinear differential equations, their analytical solution is not possible. The numerical integration, at the beginning, was encountered by problems, also, due to the lack of cheap computers and robust integration algorithms. Only after the Gear algorithm5 for the integration of stiff systems of differential equations was developed, related to the mass production of personal computers, did numeric integration become a routine procedure in kinetic investigations. Basically, numeric-integration-related investigations of the anticipated models are fitting procedures trying to adjust insufficiently known (or unknown) rate constants to simulate oscillatory dynamics as best as possible. Although looking deceptively simply, finding the set of parameters leading to chemical oscillations is a difficult task. As a result, a whole new theoretical branch of investigations based on the Clarke analysis6 is developing. Despite the subtle approaches in the mentioned investigations, the basic uncertainty related to the initially assumed model remains. One problem comes from the fact that making models is not unique. Various models with different * E-mail: [email protected].

combinations of feedback loops may simulate the oscillations.7 Another problem is related to the highly nonequilibrium conditions under which the oscillations appear. Under such circumstances a specific distribution of energy, overlooked by a purely formal approach, may determine the behavior of the process, making its dynamics even more complex and interesting. As a model system in the present investigations, the inorganic Bray-Liebhafsky8,9 (BL) oscillatory reaction is chosen. It is chosen because of the simplicity of its initial composition and interesting experimental data related to (a) the specific effects of mixing indicating nucleation-like properties of nonequilibrium transitions,10 (b) intensive changes of hydrogen bond dynamics during oscillations in the BL system recorded by NMR11 (and the Belousov-Zhabotinsky oscillator12), and (c) the possibility to influence oscillations by changing the water network either with microwave heating13-15 or by addition of large amounts of D2O.16 Together with the preliminary results introducing nonequilibrium energy distributions as an important part of the reaction mechanism,17 the mentioned peculiarities suggest that the BL reaction dynamics is more interesting that can be anticipated from the simple formal kinetic approach. The BL reaction, also, belongs to the large class of oscillatory systems containing hydrogen peroxide.3 The general scheme of the process consists of two complex reactions periodically dominating each other:8

2IO3- + 2H+ + 5H2O2 f I2 + 5O2 + 6H2O2

(1)

I2 + 5H2O2 f 2IO3- + 2H+ + 4H2O

(2)

Under the appropriate conditions, oscillations are easily observable by the periodic increase and decrease of the iodine concentration (and other involved but not so easily detectable intermediates). Following chemical oscillations in the system is more conveniently performed by monitoring the potential of the platinum electrode, which reflects changes of the redox properties of the reaction mixture. An interesting property of the general reaction scheme is that hydrogen peroxide acts both as a reducing agent in reaction 1 and as an oxidizing agent in reaction 2. The main problem for modeling is that, although thermodynamically favorable, reaction 2 is extremely slow and iodine in the presence of peroxide can stay unchanged for hours.8,18 It is not clear how process 2 with

10.1021/jp908888y  2010 American Chemical Society Published on Web 11/17/2009

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Figure 2. (a) Signal from the vessel walls with the optic probe focused on the surface. (b) First recorded Raman spectrum from the solution. Figure 1. Oscillatory evolution of the BL reaction recorded by the Pt electrode. The arrow indicates the moment where the collection of Raman spectra was started.

a higher activation energy can dominate process 1 under the BL reaction conditions. As was discussed previously,17 one possible explanation is related to a specific vibration activation of peroxide, increasing its oxidative properties. Periodic excitation of the peroxide first vibration state is indicated by Raman spectroscopy although with a large noise. Since the BL process is conducted isothermally (within 0.5 °C), a specific nonequilibrium distribution of energy is assumed as an important contribution to the reaction mechanism. The main purpose of the present work is to further investigate previous preliminary results and, if possible, find a closer correlation of the energy dynamics with the concentration oscillations. A more powerful Raman system is used in the experiments, giving a better quality of the data. Also, by using the Raman technique, a simple vibration spectrum of the investigated water solution is obtained, enabling relatively easy assignment of spectral peaks. Experimental Section The Bray-Liebhafsky oscillatory reaction was conducted in a cylindrical quartz tube of diameter d ) 18 mm. The vessel was thermostated with a circulating water bath at a temperature of 63.5 ( 0.5 °C. It was measured with an NTY84 temperature sensor placed in a glass tube for chemical isolation. To obtain the even distribution of temperature and more reproductive oscillations, the reaction mixture was stirred with the magnetic stirrer at 400 rpm. Chemical oscillations were recorded by monitoring the potential of a platinum electrode versus a Ag/ AgCl double junction reference electrode. The outer electrolytic bridge was filled with saturated K2SO4 to prevent interference of the chlorides with the BL reaction. The volume of the reaction solution was V ) 5.2 mL, and the initial composition was [KIO3]0 ) 7.2 × 10-2 mol/dm3, [H2SO4]0 ) 4.8 × 10-2 mol/ dm3, and [H2O2]0 ) 0.4 mol/dm3. Deionized water of resistivity F ) 18 MΩ cm was used for the preparation of the solutions. All chemicals were of p.a. grade from Merck. A Thermo Scientific Nicolet Almega XR Raman spectrometer with a fiber optic probe was used for obtaining the spectra. A 780 nm laser working at a maximum power of 150 mW was used for sample illumination, enabling a spectral resolution of 2 cm-1. The series of spectra were collected after 21 s of illumination with a time between recordings of 21.95 s. Results The dynamics of the BL reaction recorded by the platinum electrode is represented in Figure 1. The oscillatory regime is preceded by a “smooth” induction period, “preparing” for the oscillations. Collection of the Raman spectra started during the

induction period of the reaction indicated by the arrow in Figure 1. Before the Raman signal from the reaction solution was recorded, the signal appearing from the vessel material was initially recorded by focusing the optical probe at the surface of the quartz vessel wall. This is necessary for further processing of recorded spectra from the solution. The resulting spectrum is shown in Figure 2a together with the first recorded spectrum from the solution, Figure 2b. The intensity of the quartz peaks depends significantly on the focusing of the fiber-optic probe, and it is only intended to show the area of their contribution to the spectrum. From the first Raman spectrum of the reaction mixture (with the focus in the solution), Figure 2b, five noticeable peaks can be defined in relation to the literature data: 1629 cm-1 (bending vibration of water),19 1051 cm-1 (HSO4-),20 978 cm-1 (H2SO4),20 874 cm-1 (H2O2),21 798 cm-1 (IO3-).22 The large shoulder at the left wing of the iodate peak may be attributed to the overlapping quartz signal. For the present investigations, intended to follow the energy dynamics by monitoring peroxide vibration transitions, the most interesting region is positioned between the peroxide (874 cm-1) and iodate (798 cm-1) peaks. The literature data report for the peroxide vibration21 880 cm-1 and the iodate vibration22 805 cm-1. Since the comparison of these peak maxima in the present experiments with two independent literature sources gives a difference of about 6 cm-1 in both cases, it can be attributed to the systematic shift by the instrument itself. It is further supported by recorded spectra of pure components which confirmed that the shifted positions of the peak maxima at 874 and 798 cm-1 are not a result of molecular interactions in the observed reaction mixture. The area of interest, 874-798 cm-1, should contain a peroxide vibration transition from its ground state (0 f 1) and the first two excited vibration states (1 f 2 and 2 f 3 transitions). As is defined previously,17 taking into account anharmonicity of the peroxide vibration, they are shifted from the 0 f 1 transition at 784 cm-1 (880 cm-1) to lower energies by 23 ( 3 and 46 ( 4 cm-1, respectively, and positioned at (corrected values for the instrument shift of 6 cm-1 are given in parentheses)

ν˜ 0 ) 874 cm-1; ν˜ 1 ) 851 ( 3 cm-1; ν˜ 2 ) 828 ( 4 cm-1 (880 cm-1)

(857 cm-1)

(834 cm-1)

(3) Having the wavenumbers of the corresponding transitions, the equilibrium fractions of the molecules in excited states at 63.5 °C can be calculated according to statistical thermodynamics:23,24

f1/f0 ) e-hcν˜ 0/kT ) 2 × 10-2 f2/f0 ) e-hc(ν˜ 0+ν˜ 1)/kT ) 6 × 10-4

(4)

In eq 4 the meanings of the symbols are as usual: fi is the fraction of molecules in the ith vibration state (with energy εi ) hcV˜ i) based on the Boltzmann distribution function fi ) N0(e-εi/kT/

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deconvolution: ν˜ 1 ) 854 ( 2 cm-1, ν˜ 2 ) 826 ( 3 cm-1 prediction: ν˜ 1 ) 851 ( 3 cm-1, ν˜ 2 ) 828 ( 4 cm-1

(5)

Figure 3. Deconvolution of the quartz signal into two Gaussian peaks in the area of interest.

Figure 4. First recorded Raman spectrum from the solution and the peaks obtained by deconvolution: H2O2 (874 cm-1), [H2O2]* (853 cm-1), IO3- (799 cm-1), quartz signal simulated with two Gaussian peaks (729 and 788 cm-1).

∑ie-εi/kT) in the system containing N0 molecules, h is the Planck constant, k is the Boltzmann constant, and c is the speed of light. Looking at Figure 2 and eq 3, the expected weak signal at 828 cm-1 should be superposed to signals from quartz and iodate (Figure 2). Also, the peak at 851 cm-1 should be positioned at the right wing of the peroxide peak. To extract information from the spectra of overlapping peaks, a deconvolution process with the PeakFit software is performed. Signals corresponding to peroxide at 874 cm-1 and iodate at 798 cm-1 (Figure 2b) are represented by Voight shaped peaks (suitable for liquid samples) as well as signals from excited peroxide expected around 828 and 851 cm-1. Taking into account the contribution from quartz is enabled by simulating the signal from pure quartz in the area 900-700 cm-1 by two Gaussian peaks (suitable for solid samples) positioned at 729 and 789 cm-1, Figure 3. The result of deconvolution for the first recorded spectrum (Figure 2b) in the range 950-700 cm-1 is shown at Figure 4. As can be seen, only the peak at 853 cm-1 (without the second excitation peak at 828 cm-1) is necessary for fitting the first obtained spectrum. To eliminate the signal from the vessel walls in the subsequently recorded spectra, two Gaussian peaks representing the quartz signal are “locked” (peak parameters are fixed), and these values are imported into the software for subsequent deconvolutions. Such a procedure simulates the constant contribution of the quartz signal in all recorded spectra, since the position of the fiber-optic probe was fixed during the experiment. With this procedure of taking the quartz signal into account, all spectral noise is ascribed to four peaks of interest. The same procedure of fitting spectra with four peaks (plus two fixed quartz peaks) is applied to all 62 spectra recorded during the preoscillatory and oscillatory regions. The mean values of the peak centers of the excited peroxide molecules obtained by deconvolution of the recorded spectra and predicted by theory are

The stated errors represent the standard deviation obtained from all recorded spectra. The agreement between two groups of data supports the intention of acquiring information about peroxide excitations from the recorded spectra. The obtained peak areas Aνi are approximately taken to be proportional to the amount fi of the peroxide molecules in the corresponding vibration states νi, i.e., their concentration (Aν1 ≈ f1 ≈ [H2O2]* and Aν2 ≈ f2 ≈ [H2O2]**). The periodic appearance of excitation peaks would reflect changes in the energy flow during the reaction so that the correlation of the peak areas with the chemical oscillations is investigated. The absolute values of the peak areas are of little use because of a small signal appearing from excited molecules and fluctuations (in the mixture and instrument electronics) during the experiment. More useful data are extracted by observing the amounts of excited molecules relative to the amount of molecules in the ground state:

Aν1/Aν0 ) f1/f0 ) [H2O2]*/[H2O2] Aν2/Aν0 ) f2/f0 ) [H2O2]**/[H2O2] The results are given in Figure 5 showing the correlation among the appearances of [H2O2]**/[H2O2] and [H2O2]*/[H2O2] excited peroxide molecules (parts a and b, respectively, of Figure 5) and the reaction dynamics recorded by the platinum electrode (Figure 5c). The maximal fraction of peroxide molecules in the first excited state of [H2O2]* is achieved during the smooth period between the chemical oscillations. A minimal value of [H2O2]*/ [H2O2] ≈ 0.02 corresponds satisfactorily with the equilibrium value (eq 4), and a maximal value of ∼0.11 corresponds to a ∼5× higher population of excited molecules. From Figure 5a, the maximal fraction of [H2O2]** molecules does not coincide with the maximum of the first excitation, and it can be more satisfactorily correlated with the maximum of the platinum electrode potential. A maximum value of [H2O2]**/[H2O2] molecules of ∼0.04 corresponds to a ∼60× higher population from the equilibrium value (eq 4). Taking into account that the temperature of the mixture is kept constant at 63.5 ( 0.5 °C, the obtained results strongly support the existence of a nonequilibrium distribution of energy preceding the appearance of every “chemical” oscillation. Despite considerable noise, cal-

Figure 5. Time correlation among (a) the fraction of [H2O2]** excited peroxide molecules, (b) the fraction of [H2O2]* excited peroxide molecules, and (c) the potential of the platinum electrode.

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culated errors confirm that the periodicity extracted from the spectral data is correlated with the concentration oscillations and that the appearance of [H2O2]** molecules is phase shifted relative to the emergence of [H2O2]*. Discussion In the present experiments the dynamics of energy distribution during the oscillatory BL reaction is investigated following the population of excited peroxide molecules during the course of the reaction. All experiments are performed under mechanical stirring to minimize the local increase of temperature and for a better reproducibility of the chemical oscillations. Excited states of peroxide are detected using Raman spectroscopy with a 2 cm-1 resolution of the recorded spectra. Higher spectral resolution together with the elimination of the signal from the vessel walls enabled a more detailed analysis of the spectra. In addition to the increased amount of peroxide in the first vibration state,17 an excessive number of molecules in the second vibration state is monitored. As can be seen from Figure 5, before every oscillation recorded by the platinum electrode, a considerable rearrangement of energy appears in the system. Part of this energy provokes vibration excitation of peroxide and possibly enough change of its reactivity to redirect the course of the reaction. This may underline the very specific role it plays in the reaction mechanism and, also, explain the unusually large number of chemical oscillators containing hydrogen peroxide.3 Although identification of the excited peroxide molecules is dependent on the deconvolution procedure, good agreement of the obtained and predicted wavenumbers for the vibration transitions in peroxide (eq 5) supports the initial assumption of their existence. Expressing the relative amounts of excited molecules (relative to the amount of peroxide in the ground state) partially eliminated the noise in the signal due to the fluctuations in the reaction mixture and the instrument itself. Figure 5 shows that the uncertainty of the data is not too high, and the results reflect changes of the energy dynamics during the course of the reaction. This and good periodicity of the energy changes, well correlated with the concentration oscillations, confirm that they are a result of the reaction mechanism rather than random fluctuations of the experimental conditions and data processing. As is shown (eq 4) excitation of peroxide molecules significantly exceeds the equilibrium values. Taking into account that the bulk temperature of the mixture is kept constant during the reaction, the existence of a specific nonequlibrim distribution of energy as a part of the reaction mechanism is strongly supported. Excitation due to the high local temperature increase seems improbable for two reasons. One is that in the present experiments intensive stirring of the mixture is performed (5.2 mL of the solution is mixed at 400 rpm). The other reason is that excitation of the first and second peroxide vibration states is phase shifted. If they are a result of temperature only, excitations should occur simultaneously. As is more thoroughly discussed in a previous paper,17 the excessive nonequilibrium excitations of the lowest vibration states are predicted by Frohlich25,26 for isothermal systems exchanging energy with the surroundings and pumped internally with the energy (released from chemical reactions) over some threshold rate. The excess energy in the established stationary states can then be channeled to the lowest vibration states of the molecules with a selective change of the molecules’ reactivity. Numerical simulations of the Frochlich mechanism by Mesquita and co-workers27,28 based on the NESOM (nonequilibrium statistical operator method29) formalism confirmed

Stanisavljev very fast (on the subpicosecond time scale) excessive excitation of the five lowest vibration modes of the investigated biopolymers. If applicable to the BL reaction, such a process should be accompanied by the active involvement of liquid water which is present in a great excess over all components and inevitably stores the largest part of the energy released in chemical reactions. That such activity of the hydrogen-bonded water network may be an important part of the oscillatory mechanism shows large changes of its dynamics recorded in the Belousov-Zhabotinsky12 and Bray-Liebhafsky oscillatory reactions.11 It is further supported by a considerable influence of D2O on the reaction mechanism16 which could not be entirely ascribed to the ordinary isotopic effects as well as the microwave-heated reaction solution14 with the forced changes of the hydrogen bond dynamics. The present experiments further strengthen the assumption of an energy flow as an important part of the BL reaction mechanism. According to the obtained results, stationary states producing excited molecules periodically change in time. Well before chemical oscillation is initiated, the energy is rearranged so that the first vibration level of peroxide is detected, followed by the excitation of the second vibration level. As is discussed previously,17 excitation of higher vibration states, with the assumed Frohlich-like mechanism, is probably not relevant because of breaking of the hydrogen-bonded network, transmitting too large an amount of energy. All this reveals that even in the smooth reaction periods with the stationary intermediate concentrations a severe energy dynamics appears, providing (possible) conditions for triggering concentration oscillations. Although the way energy is distributed to the molecules in the solution is unknown, the excessive appearance of excited peroxide molecules may help in better understanding the switching mechanism between reactions 1 and 2 and establishing more detailed reaction models. As is explained,3,7 the most intriguing concept in establishing reaction models is the existence of autocatalytic steps of the form A + 2X f 3X. Although in the formal kinetic approach the energetics of such processes are not concerned, this is of critical significance to the models’ reliability. In that sense, introduced nonequilibrium energy rearrangements may be involved in the evolution of autocatalysis, forming a new kind of feedback in evolving chemical oscillations. A more precise relation to the reaction mechanism should be established by further research. Although the results obtained in the present work strengthen previously introduced assumptions,17 important differences and outcomes should be stressed. Because in both experiments the deconvolution procedure is used in data processing, the repeatability of the results with the more sensitive Raman system increased considerably the reliability of assumed nonequilibrium energy distributions in the BL system. Of inherent importance in the present experiments is the existence of the second excitation state of peroxide. Consistency of the excited peak positions with the theoretical predictions additionally confirmed correct assignments of the vibration signals and possibly the quite specific role peroxide plays in the reaction mechanism. The more precise data in the present experiments showed that the flow of energy has its own dynamic structure, producing successive excitation of two excited peroxide states. In addition, the quality of the results is improved due to the efficient stirring of the mixture, which eliminated the uncertainty related to the temperature inhomogeneities. A more general consequence of the obtained results is the imposed controlling function of the liquid water network, which

Energy Dynamics in the Bray-Liebhafsky Reaction may be of high importance in living organisms and other nonequilibrium systems. Conclusion Using Raman spectroscopy of a better resolution, a more precise analysis of vibration spectra during the oscillatory and nonoscillatory evolution regimes of the Bray-Liebhafsky reaction is performed. Periodic excitation of the first and second vibration levels of peroxide preceding chemical oscillations is detected. The appearances of excited molecules, high above the expected isothermal equilibrium populations, suggest severe changes in the energy dynamics. Good correlation between chemical oscillations and energy rearrangements shows that they are an intrinsic part of the reaction mechanism. This may provide better insight into the process of the switching mechanism between different kinetic pathways and triggering chemical oscillations. The existence of such mechanisms is of more general significance because it anticipates the important role of water which is present in high excess and due to the high health capacity practically should control all of the energy rearrangements in the system. Acknowledgment. This work is supported by the Ministry of Science and Environmental Protection of Serbia under Project Number 142025. I express special gratitude to Danica Bajuk for assistance in collecting the Raman spectra. References and Notes (1) Glansdorf, P.; Prigogine, I. Thermodynamic Theory of Structure Stability and Fluctuations; Wiley: New York, 1971. (2) Prigogine, I. Therodynamique des Processus IrreVersibles; Desoer: Liege, Belgium, 1947. (3) Epstein I. R.; Pojman, J. A. Introduction to Nonlinear Chemical Dynamics; Oxford University Press: New York, 1998. (4) Goldbeter, A. Biochemical Oscillations and Cellular Rhythms; Cambridge University Press: Cambridge, U.K., 2002.

J. Phys. Chem. A, Vol. 114, No. 2, 2010 729 (5) Gear, C. W. Numerical Initial Value Problems in Ordinary Differential Equations; Prentice Hall: Upper Saddle River, NJ, 1971. (6) Clarke, B. L. AdVances in Chemical Physics; Wiley: New York, 1980. (7) Eiswirth, M.; Freund, A.; Ross, J. AdV. Chem. Phys. 1991, 80, 127. (8) Bray, W. C. J. Am. Chem. Soc. 1921, 43, 1262. (9) Liebhafsky, H. A. J. Am. Chem. Soc. 1931, 53, 2074. (10) Roux, J. C.; De Kepper, P.; Boissonade, J. Phys. Lett. 1983, 97A, 168. (11) Stanisavljev, D.; Begoviæ, N.; Z´eujoviæc, Z.; Vuc`eeliæc, D.; Bac`eiæc, G. J. Phys. Chem. A 1998, 102, 6883. (12) Hansen, A. W.; Ruoff, P. J. Phys. Chem. 1989, 93, 264. (13) Stanisavljev, D.; Djordjeviæc, A. R.; Likar-Smiljaniæc, V. D. Chem. Phys. Lett. 2005, 412, 420. (14) Stanisavljev, D.; Djordjeviæc, A. R.; Likar-Smiljaniæc, V. D. Chem. Phys. Lett. 2006, 423, 59. (15) Stanisavljev, D. R.; Djordjeviæc, A. R.; Likar-Smiljaniæc, V. D. ChemPhysChem 2004, 5, 140–144. (16) Stanisavljev, D.; Begoviæc, N.; Vukojeviæc, V. J. Phys. Chem. A 1998, 102, 6887. (17) Stanisavljev, R. D.; Dramicanin, D. M. J. Phys. Chem. A 2007, 111, 7703. (18) Furrow, S. J. Phys. Chem. 1987, 91, 2129. (19) Marechal, Y. The Hydrogen Bond and the Water Molecule; Elsevier: Amsterdam, 2007. (20) Gillespie, J. R.; Robinson, A. E. Can. J. Chem. 1962, 40, 644. (21) Giguere, P. A.; Srinivasan, K. K. T. J. Raman Spectrosc. 1974, 2, 125. (22) Nakamoto, K. Infrared and Raman Spectra of Inorganic and Coordination Compounds; Wiley: Hoboken, NJ, 2009. (23) Alberty, A. R. Physical Chemistry; Wiley: New York, 1987. (24) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1976. (25) Frohlich, H. Int. J. Quantum Chem. 1968, 2, 641. (26) Frohlich, H. AdV. Electron. Electron Phys. 1980, 2, 85. (27) Mesquita, V. M.; Vasconcellos, R. A.; Luzzi, R. Phys. ReV. E 1993, 48, 4049. (28) Mesquita, V. M.; Vasconcellos, R. A.; Luzzi, R. Phys. Lett. A 1998, 238, 206. (29) Luzzi, R.; Vasconcellos, R. A.; Ramos, J. G. PredictiVe Statistical Mechanics: A Nonequilibrium Ensemble Formalism; Kluwer: Dordrecht, The Netherlands, 2002.

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