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J. Phys. Chem. 1992, 96, 9298-9301
(10) Bebelaar, D. Chem. Phys. 1974,3, 206. (1 1) Shida, T. Electronic Absorption Spectra of Radical Ions; Elsevier: New York, 1988. (12) Asmus, K.-D.; Fendler, J. H. J. Phys. Chem. 1968, 72,4285. (13) Bansal, K. M.;Fessenden, R. W. J. Phys. Chem. 1976, 80, 1743. (14) Hatano, A., Matuoka, S., Chapter Eds. C R C H a n d h k of Radiation Chemistry; Tabata, Y . , Ed.; CRC: Boca Raton, FL, 1990; pp 190-202. (1 5) Ionization Potential and Appearance Potential Measurements, 1971-1981; Report NSRDS-NBS 71; U. S. Government Printing Office: Washington, DC. (16) Parker, V. D. J . Am. Chem. Soc. 1976, 98,98. (17) Friedman, H. L.; Krishnan, C. V. Water. A ComprehensiveTreatise. In Aqueous Solutionr of Simple Electrolytes; Franks, F., Ed.; Plenum: New York-London, 1973; p 56. (18) Because G values vary by a factor of nearly 20, this trend is clear even with the uncertainty in the values of the extinction coefficients of the aromatic
radical cations. The extinction coefficients were estimated from values in the literature. We assumed that the oscillator strength was approximatelyconstant for a particular cation in the different solvents, and thus the maximum absorption coefficient in our solutions could be estimated using the widths of the absorption band. (19) Mataga, N.;Asahi,T.; Kanda, Y.; Okada, T. Chem. Phys. 1988,127, 246. (20) Wang, Y.; Tria, J. J.; Dorfman, L. M. J . Phys. Chem. 1979,83 (15). 1946. (21) Borovkova, V. A.; Kirynkhin, Yu. I.; Sinitsyna, Z. A.; Romashow, L. V.; Bagdasar'yan, Kh. S. High Energy Chem. 1984,88 (IS), 192. (22) KirynLhin, Yu. I.; Borovkova, V. A.; Sitsyna, Z. A,; Bagdasar'yan, Kh. S. High Energy Chem. 1979,13 (6). 428. (23) Gschwind, R.; Haselbach, E. Helv. Chim. Acta 1979, 62 (4), 941. (24) Reichardt, C. Solvent and Solvent Effects in Organic Chemistry; VCH: Weinheim, 1988; Table A-I.
Comparative Behavior of the Autocatalytic Mn0JH2C20, Reactlon In Open and Closed Systems V. Pimienta, D.Lavabre, G.Levy, and J. C. Micbeau* Laboratoire des IMRCP, URA au CNRS 470. UniversitL Paul Sabatier, F-31062 Toulouse, France (Received: March 30, 1992; In Final Form: August 3, 1992)
If the rate equation of a chemical reaction can be expressed in terms of a single variable, the steady-state conditions in a CSTR can be predicted from the batch kinetia. This does not require detailed knowledge of the reaction mechanism. However, if the reaction comprises several elementary steps, deduction is not possible since the rate depends on several independent concentrations. Experimentally useful predictions can nevertheless be made in certain situations. For example, the presence of an isosbestic point in the UV/visible spectrum of a reaction mixture is frequently indicative of a single-variable process. The permanganateoxalic acid reaction is a convenient system for testing the method since it exhibits an isosbestic point in batch and its shows bistability in a CSTR for a limited range of oxalic acid concentrations. By using the rate data for two different oxalic acid concentrations, one can predict that bistability will be exhibited for one but not the other. The prediction is confirmed by experiments using a CSTR.
Introduction In order to design new bistable chemical systems, we evaluated autocatalytic behavior from batch kinetics. This approach has the merit of speed and simplicity. However, to find out whether a particular system will exhibit bistability in a continuous stirred tank reactor (CSTR), appropriate calculations are on the simplifying assumption that the course of the reaction can be defined in terms of a single differential equation, which only depends on a single variable. A simple graphical method can be employed to deduce the steady-state curve and the multiplicity of a reaction in a CSTR starting from batch kinetics. However, if there are several elementary steps, the rate of the reaction will normally depend on several concentrations, and the graphical method is no longer applicable. In this case, bistability cannot be predicted r e l i a b l ~ . ~ We show here that, in some experimental situations, useful predictions can be made even though the reaction mechanism is not known in detail. In the case of the autocatalytic reduction of the Mn04- ion by oxalic acid, multiplicity in a CSTR can be predicted from batch experiments. Noyes-Epsteia S i V a r i a b l e Graphical Method4 The single diffmntial equation d e f h g the behavior in a CSTR is
This equation includes two terms: (1) the flux term, which is a linear function of the concentration [A] of reactant A, and (2) the chemical term vc, which is the algebraic sum of the rates of all chemical processes involving species A in the CSTR. T is the residence time in the CSTR and [AIothe concentration of reactant A in the input. Steady states are reached when the flux term exactly counterbalances the chemical term.s This gives
dA/dt = 0 (2) The chemical rate vc is thus expressed as a sole function of [A] and is the same in both types of reactor. It is plotted as a function of the residual reactant [AI0 - [A]. The flux straight line, vF= ( ~ / T ) ( [ A] ~[A]), is traced on the same diagram. ( l / r ) is the slope. The steady states (progress and residence time) are indicated by the intersedim. If the accelerationof an autocatalytic reaction is sufficiently marked, the rate curve is concave, and it will intersect the flux line at three points for certain values of residence time. This zone corresponds to the residence time domain where bistability will be observed. It is bounded by the tangents (dotted lines 2 and 4 in Figure 1) to the curve passing through the origin.6.' At low residence times, the intersections define the flux branch, and at high residence times, it defines the thermodynamic branch.* Although rapid and convenient, this graphical method has limited applicability. For multistep reactions, the chemical rate will be a function of several independent concentrations. Thus uc = V(XI,..X") (3) where Xi are the concentrations of species involved in the reaction mechanism. The rates are not the same in the two types of reactor (CSTR and batch). We show, however, that the observation of an isosbestic point (or linearly related spectra9)enables reduction of the number of independent variables to approximate a single variable
Reduction of the Nomber of Variables In a CSTR, the course of a reaction involving various elementary steps is defined by a series of n differential equations of the type dXi/dr = fi(Xi) + vci(X,,..Xn) (4)
0022-3654/92/2096-9298so3.Q~ f 0 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 9299
The Autocatalytic Mn04-/H2C204Reaction 1.0
I
/
1
Figure 1. Graphical representation of the chemical rate u as a function of the residual reactant [AIo- [A]. The flux term is represented by the straight line passing through the origin. The point(s) of intersection rep”t(s) the steady states in a CSTR. (1) Flux branch; (3) butability zone; (5) thermodynamic branch.
wheref, and ua are the flux and chemical rate terms, respectively. The equations of conservation can be used to reduce the number of variables. Furthermore, in the presence of an isosbestic point, i.e., a point at which the UV/visible absorbance does not change with time, the additivity of the Beer-Lambert’s law gives Absim/l ... tJn (5)
+ +
where A& is the absorbance at the isosbestic wavelength, 1 the optical path length of the reactor, and ti the molar extinction coefficient of species j of concentration Xiat the wavelength of the isosbestic point. The relationship 5 hereafter noted as the isosbestic point equation is similar to a conservation equation. In consequence, if eq 5 is independent, the presence of an isosbestic point allows the number of variables to be reduced by a unit. The observation of isosbestic points (or internal linearity) in a reaction system should thus be borne in mind when a single-variableapproximation is required. The isosbestic point does not need to be present throughout the duration of the reaction, although in this case variable reduction is restricted to the phases of the reaction in which the isosbestic point (or internal linearity) is present. These considerationscan be readily illustrated in systems involving only two or three species. (a) Reactions Involving Only Two Species A and B. These reactions are always single-variable systems as imposed by the conservation equation [A10 +
P I 0
= [AI + [BI
(6)
relating the concentrations of the two species. The isosbestic point stems uniquely from the equality o f t (tA = %) and d m not reveal any new constraints on the concentrations. (b) Reactions Involving Only Three Species A, B, and C. Two main cases must be taken into account: (i) If one can write two independent conservation equations, for instance, [CI + [AI = [A10 (74 [CI + P I = [Blo (7b) the system reduces to a single variable. As in the previous case, the isosbestic point stems solely from the relationship between the (q =
+
(ii) On the other hand, if one can write only one conservation equation, for instance, [A10 + P I 0 + [Clo I[AI + P I + [CI (8) an isosbestic point can have two meanings:
(i) If there is a wavelength where tA = tg = CC, the isosbestic point, is observed at this wavelength, no constraint is impaeed on the concentrations, and so the system cannot be reduced to a single variable. This situation is, however, highly improbable as it corresponds to the intersection of three independent W spectra. (ii) If such a wavelength does not exist, the presence of an isosbtic point imposes a relationship between the concentrations of the remaining species, and from eqs 5 and 8 it appears that [B](tB- PA) + [C](cc - tA) vanishes at the isosbestic point and [B]/[C] = ( t -~Q)/CB - (A)). This implies that the ratio [B]/[C] remains constant throughout the reaction. This situation is characteristic of two competitivereactions from the same substrate A. The isosbestic point appears at the wavelength where the above relationship applies. The relationship between [B] and [C] thus reduces such a reaction to a single variable system. (c) Systems lmohg More Species For systems involving more species, the situation is more complex. Nevertheless, if the isosbestic point equation is independent of the conservation equations, the number of variables in the system can be reduced by 1, and in favorable cases, to a singlevariable situation. This only applies, however, to the period during which the isosbestic point is observed.
Reaction between the MnOl Ion and Oxalic Acid in the Presence of Sulfuric Acid The oxidoreduction equation of 1 / 2 5 stoichiometry is 2Mn04- + 6H+,q + 5HzC204= 2Mn2+ + 8H20 + l o c o 2 (1) This reaction, which was first studied kinetically in 1866,” has the following characteristics: 1. It is auto~atalytic.’~ 2. The strength of the autocatalysis depends on the initial molar ratio [MnOyl0/ [H2C2O4I0, since the reduction of permanganate is inhibited by a limited excess of oxalic acid.I5 In a CSTR, a monostable situation is created with excess oxalic acid, whereas bistable behavior16is produced with a stoichiometric molar ratio of the two reactants. 3. In a batch reactor, an isosbestic point is observed at A = 457 nm in the UV/visible spectrum of the reaction mixture at the start of the reaction. The specific absorption of MnOC ions (490 < A < 570 nm) is replaced by a transient absorbance at 300 nm as the reaction proceeds. The solution may also contain significant concentrations of species which contribute to kinetic behavior but do not absorb significantly at the wavelengths which were studied. However, this peak is likely to be due to one or more intermediates whose structure is not yet known in detail, which we will refer to as “Int” hereafter.” The isosbestic point does not persist until the end of the reaction, which is indicative of a much more complicated reaction mechanism. Nonetheless, during the period when the isosbestic point is observed, it can be assumed’* (i) that Int is the only other manganese species present in significant concentration along with permanganate and, (ii) that the overall reaction behaves as if MnOc Int (Figure 2). In the presence of excess oxalic acid (initial molar ratio [Mn04-]o/[HzC,04]o= 1/16-67;condition I), the isosbestic point is seen until 65% of the initial permanganate has been used up. During this initial phase, the reaction behaves as a singlevariable system: the Mn04-ion is transformed into intermediate Int,whose degradation is negligible. Conservation of Mn gives [Mn04-] + [Int] = [MnO4-Io (9) The system can thus be completely defined if [Mn04-] is known. At the end of the reaction, the isosbestic point disappears because degradation of Int predominates. Equation 9 is no longer valid. For a stoichiometricinitial molar ratio [Mn04-]0/ [H2C20410 = 1/23 (condition 11), the reaction is faster than in condition I as there is little inhibition from oxalic acid (Figure 3). Analysis of the three-dimensional spectra in condition I1 showed that the isosbestic point remained until 44% of the initial permanganate had been used up, After this period, both absorbances of Mn04and Int fell rapidly.
-
Pimienta et al.
9300 The Journal of Physical Chemistry, Vol. 96, No. 23, I992
L Oo
700 o
X
i
-
Figure 4. (A) Diagram of Y vs X,the rate was normalized to the maximum rate observed during the run (dAbsSa/dt)-: Y = dAbssa/dt/ (dAbsSa/dt)- and the residual concentration of permanganate at the initial concentration; X = ([MnO4-Io- [Mn04-])/[Mn0,10. The zones where the isosbestic point is observed are marked by full lines. The flux lines are of equation Y = (1/7)X by varying 7 . (B) Curves of steady states in CSTR (Abssa vs 7 , monostability): (---) prediction from batch kinetics; (0) experimental observations. The arrows indicate when the transient isosbestic point disappears.
Figure 2. Three-dimensional spectra (Abs, A, t ) recorded in a batch reactor. Experimental conditions I (excess oxalic acid); T = 25 "C. ( ~ ( M n 0 ~=- )1390 ~ ~ M-' cm-'). Insert: persistence of isosbestic point at 457 nm at start of reaction. LO
0
E a O.5
tau
40
Figure 5. (A) Diagram of Y vs X. Initial molar ratio [Mn04-lo/ [H2C20?]p= 1 / 2 5 (B) Curves of steady states in CSTR ((AbsSa vs 7 , bistability): (---) prediction from batch kinetics; (0)experimental observations. The arrows indicate when the transient isosbestic point disappears.
a
0.0
time(s) Figure 3. Ahsa during reduction of Mn0; ions concentration by oxalic acid in acidic aqueous solution. Curve I: Experimental condition I, excess oxalic acid, initial molar ratio [Mn04-]o/ [H2C204]0 = 1/ 16.67. Curve 11: Experimental condition 11, stoichiometric molar ratio [MnO;]o/[H2C204]o = 1 / 2 5 The arrows indicate when the transient isosbestic point disappears.
Prediction of Steady-State Conditions in a C3TR From the kinetics recorded in a batch reactor, the rate curves were calculated as a function of reaction progress. In the experimental conditions I (excess oxalic acid), the rate curve is convex. There is no more than one point of intersection with the flux line, indicating that the reaction will probably be monostable in a CSTR. We in fact observed monostable behavior in a CSTR, and the steady states were predicted accurately for short residence times. This was expected as all the corresponding points lay in the zone where the isosbestic point is observed (Figure 4). Outside this zone, the discrepancy between the predicted and observed steady states can be explained by the nonapplicability of the graphical method. All the points lying over the thermodynamic branch beyond the zone of persistence of the isosbestic point cannot be treated in a rigorous way using the single-variable assumption. This discrepancy does not, however, affect the prediction of monostability. In the experimental condition I1 (stoichiometricratio), the rate curve is concave with an inflection point. This is indicative of multiple steady states at certain values of residence time.19*20 In fact there may be three points of intersection. The first two lie inside the zone of observation of the isosbestic point, while the third lies outside it. The first two can be employed to calculate
the flux and unstable branches. An estimate of the thermodynamic branch can be obtained from the third point. This reaction will probably exhibit bistability in a CSTR (Figure 5 ) . Bistable behavior was in fact observed, and the steady states were well predicted over both the flux and unstable branches, Le., at short residence times. This was expected as the corresponding points all lay within the zone of observation of the isosbestic point. On the thermodynamic branch, the discrepancy between the predicted and observed steady states can be accounted for by the nonapplicability of the graphical method, although it does not affect the prediction of bistability. It can be seen in Figure 5 that part of the unstable branch correspondsto the zone of persistence of the isosbestic point. The position of this part can thus be predicted accurately, thereby indicating the underlying bistability. Conclusion We show here the interest of multiwavelength UV/visible spectroscopy for study of nonlinear chemical dynamics. For the reduction of permanganate by oxalic acid, bistable behavior in a CSTR could be predicted, using a simple graphical method, from the kinetics observed in a batch reactor. Although the reaction mechanism has not be completely elucidated, bistable behavior can still be predicted: (a) since the presence of an isosbestic point enables assumption of a singlevariable situation, and (b) because this isosbestic point is present during the phase of the reaction when autocatalysis is well under way. Transient isosbestic points may serve as eye-catching indicator of possible unique variable behavior. If there is no visible isosbestic point, internal lineary can be tested by any convenient method?1*22 Experimental Part 1. Preparation of the Solutions. KMn04 (MW = 158.03), H2S04(MW = 98.07), and H2C2O4, 2H20 (MW = 126.07) were analytical grade reagents. Water was double distilled. Aqueous solutions of the appropriate concentrations were made up immediately before use.
J. Phys. Chem. 1992,96,9301-9304
2. CSTR. ( a ) Pumps. The pump system consisted of two identical Bishoff HPLC pumps. The flow rates could be adjusted over a range of 10 p L m i d to 5 mL m i d . (6)Reactor. The body of the reactor consisted of a quartz cylinder closed at both ends with Teflon blocks. The lower Teflon block contained the injectors, while the upper block was slightly conical to enable escape of bubbles. A micromagnetic stirrer was placed in the cylinder. The speed of stirring was controlled by a stepper motor. The setup was placed in a thennostated chamber inside a diodearray spectrophotometer (HP 8451). The internal volume of the reactor was 2.1 mL. The absorbance of the reacting solution was recorded through a 0.5 mm wide calibrated slit, giving an overall optical path length of 1.1 cm. Pure water was used as reference. 3. Kinefics in h t c h Reactor. Measurements were made with the same solutions in the same reactor used as batch or CSTR. One pump fed in a solution of permanganate [KMn04] = 1.09 M and [H2S04]= 0.54 M. The other pump fed in a x solution of oxalic acid (condition I, [H2C204]= 1.82X lo-’ M; condition 11, [H2C204]= 2.73 X M. The solutions were pumped in at the maximum flow rate (5 mL/min for both pumps). The pumps were switched off and the UV/visible spectra d e d . The initial concentrations in the reactor were as follows: [KMnO4Io = 5.45 X lo4 M; [H2S04]o= 0.27 M. I, [H2C204]0 M. = 9.10 X lo-’ M; 11, [H2CZ04]0= 1.37 X 4Det”m ’ tionofSteady§$atesintbecsTRThetwopumps operated at the same flow rate D (mL/min). The residence time 7 (s) is expressed by the relationship 7
= 60 X 2.1/2D
The absorbance at 560 nm was recorded until a steady state was reached. This took up to 207. To describe the mne of bistability, 7 was reduced from the thermodynamic to the flux branch; 7 was then increased from the flux to the thermodynamic branch. The two transitions from one branch to another were not observed at the same values of 7 .
9301
5. Data Treatment. The experimental data (Abs versus f ) w e n transferred from the HP 8451 spectrophotometerto a HP 9OOO Series 330 workstation running under UNIX. The software was written in C. Smoothing and derivation. Both operations were carried out using the Savitsky+olay’s alg~rithm?~ with a smoothing window width of 1 1 points and a third degree polynomial. Registry No. MnO,-, 14333-13-2; oxalic acid, 144-62-7.
References pad Notes (1) Lin, K. F. Can. J. Chem. Eng. 1979,57,476. (2)Lin, K. F. Chem. Eng. Sci. 1981, 36(9), 1447. (3) A typical example is the photolysisof thioxanthone in aerated toluene. Bistabdity was predicted from the accelerating nature of batch kinetics, but in fact monostable behavior was always okperved in a CSTR. (4)Noyes, R. M.; Epstein, I. R. J. Phys. Chem. 1983,87,2700. ( 5 ) Eptein, I. R. J. Phys. Chem. 1984,88, 187. (6)Matsuura, T.; Kato, M. Chem. Eng. Sci. 1967,22, 171. (7)Fazekas, T.;Mrakavova, M.; Nagy, A.; Olexova, A,; Treindl,L. React. Kinet. Catal. Lett. 1990,42, 181. (8) de Kepper, P.; Eptein, I. R.; Kustin, K. J. Am. Chem. Soc. 1981,103,
6121. (9)Brynestad, J.; Smith, G. P. J. Phys. Chem. 1968,72,296. (10)Polster, J.; Lachman, H. Spectrometric Titration; VCH Verlagegesellschaft: Weinheim, Germany, 1989;ISBN 3-527-26436-1. (11) Cohen, M. D.; Fischer, E. J . Chem. Soc. 1962,3044. (12)Coleman, J. S.;Varga, L. P.; Mastin, S.H. Inorg. Chem. 1970,9, 1015. (13) H a W , A. V.; EEson, W. Phifar. Trans.R. Soc. London 1866,156,
193. (14)Launer, H. F. J . Am. Chem. Soc. 1932,54,2597. (15) Malcolm, J. M.; Noyes, R. M. J. Am. Chem. Soc. 1952,74,2769. (16)(a) Scott, S.K. Acc. Chem. Res. 1987,20,186. (b) Gray, P.; Scott, S.K. Ber. Bunsenges. Phys. Chem. 1986,90,985. (17) Ganapathisubramanian,N.J. Phys. Chem. 1988, 92,414-7. (18) Adler, S.J.; Noyes, R. M. J. Am. Chem. Soc. 1955,77,2036-42. (19) Reckley, J. S.;Showalter, K. J. Am. Chem. Sa.1981,103,7012-3. (20)de Kepper, P.; Ouyang, Q.;Dulos, E. In Nonequifibrium Dynamics in Chemical Systems; Vidal, C., Pacault, A., E&; Springer: New York, 1984; PP 44-9. (21) Ainsworth, S.J . Phys. Chem. 1961,65, 1968. (22)Hugus, Jr., 2.2.;El-Awadi, A. A. J . Phys. Chem. 1971,75,2954. (23) Savitzky, A.; Golay, M. J. E. Anal. Chem. 1984,36, 1627.
Rate Constants for the Reactions of the Hydroxyl Radical with Several Partially Fluorinated Etherst Z. Zhang, R. D.Saini, M. J. Kurylo, and R. E. H i e * Chemical Kinetics and Thermodynamics Division, National Institute of Standards and Technology. Gaithersburg, Maryland 20899 (Received: April 8, 1992: In Final Form: August 17, 1992)
We have measured rate constants for the reactions of the hydroxyl radical, OH, with a number of fluorinated ethers. These ethers and their calculated atmospheric lifetimes for removal by OH (in years), estimated relative to CH3CC13,are as follows: CF3-0-CH3, 3.0; CF3-O-CHF2, 19; CHFz-O-CHF2, 2.6; CFpCH2+CHp, 0.1; CFoCHz-O-CHF2, 5.2; CyClOCFZCHFCF2-0, 26;~yclo-(CF2)3-0,>330.
Iatroduction Because of the concerns regarding the role of chlorofluorocarbons (CFCs) in ozone depletion and greenhouse warming, partially fluorinated ethers are now being considered as CFC replacements in industrial applications. Since these compounds contain neither chlorine nor bromine, they do not contribute to ozone depletion, but still they must be evaluated as possible greenhouse gases. The assessment of the greenhouse warming potential of any compound requires an accurate determination of its atmospheric lifetime, which, for many species, is dictated largely by the rate of reaction with the hydroxyl (OH) radical in the troposphere. Presently, there is very little experimental information on the OH reactivity of fluorinated ethers. We have Contribution of the National Institute of Standards and Technology.
therefore measwed the rate constantsfor the reactions of OH with a representative group of hydro fluoro ethers and have used the results to estimate their tropospheric lifetimes by means of a comparison with the reactivity of CCl3CH3. The rate constant data also provide preliminary information on the effects of fluorine substitution on OH/ether reactivities, which may help in predicting the rate constants for reactions of other hydro fluoro ethers with OH.
Experimental Section Rate measurements were conducted at 296 K by the flash photolysis resonance fluorescence technique.’-3 A pyrex reaction cell (100 cm3 volume) was used in the flow mode. Reaction mixtures containing the hydro fluoro ether, H20, and Ar were prepared manometrically in glass storage bulbs and slowly flowed
This article not subject to US.Copyright. Published 1992 by the American Chemical Society
