J. Phys. Chem. 1995,99, 14365-14371
14365
Kinetic Modeling of the KMnO&IzC2O&IzS04 Reaction: Origin of the Bistability in a CSTR V. Rmienta, D. Lavabre, G. Levy, and J. C. Micheau" Laboratoire des IMRCP, URA au CNRS 470, Universitt?Paul Sabatier, F-31062 Toulouse, France Received: May 8, 1995; In Final Form: July 18, 1995@ The mechanism of the reduction of permanganate by oxalic acid in sulfuric acid medium was completely described by a model incorporating the specific reactivities of permanganate and of various Mn(II1) and Mn(1V) reaction intermediates. It involved 14 steps including 8 equilibria (dissociation of the two diacids and complexation of Mn(III) intermediates by oxalate ions and of Mn(IV) by molecular oxalic acid). Numerical values of the parameters for this model were optimized from the fitting of a series of independent kinetic experiments carried out under conditions designed to display the main dynamic characteristics of the evolution of permanganate and intermediates. The predictive power of this model was then tested. It was found to simulate accurately the times of half-disappearance of permanganate and the intermediates as a function of the initial concentration of oxalic and sulfuric acids. The model was also able to account for bistability in a CSTR for oxalic acid concentrations close to the stoichiometric ratio. Bistability was shown to stem from a coupling between the autocatalytic reduction of permanganate by a complex of Mn(II) and a substrate inhibition like effect, namely an attenuation of the inhibition of the reduction of Mn(II1) toward the end of the reaction when the initial oxalic acid was almost completely consumed. The width of the hysteresis loop, the range of residence times, and the corresponding initial concentrations were also in quantitative agreement with the prediction of the model.
I. Introduction Although the permanganate/oxalic acid/sulfuric acid reaction was first described more than a century ago,' the recent discovery of its bistable behavior in a CSTR has renewed interest in the reaction m e ~ h a n i s m . ~ , ~ The overall reaction can be represented by the following equation: 2Mn0,-
+ 6H30+ + 5H2C,04 =
2Mn2+ -I- 14H20 -I-10C0, We carried out a study of this reaction over the range of initial concentrations of sulfuric acid, 0.02 < [H2S04]0 < 0.3 mobL-', and oxalic acid, 1.5 x < [H2C204]0 < 0.2 mol*L-', in which the most interesting dynamic effects and bistable behavior in a CSTR can be observed. In a preliminary study of the kinetics in a batch reactor, we observed an acceleration of the decay of permanganate as a function of initial sulfuric acid concentration along with a more complex effect as a function of initial oxalic acid concentration: (i) acceleration at high [H~C204]0,and (ii) inhibition at low [H2C20410. The dynamic behavior in a CSTR also depends on [H2C204]0, the reaction being monostable at high [H2C204]0 and bistable at low [H2C204]0. In the present study, we carried out a detailed kinetic study of the permanganate/oxalic acid reaction in sulfuric acid medium. Our objective was to establish a quantitative model based on a chemically realistic mechanism to describe the main dynamic effects and gain further understanding of the bistable behavior in a CSTR.
11. Experimental Study of the Reaction The kinetics of this autocatalytic reaction (the Mn2+ ion is the autocatalyst) can be monitored by UV/visible spectrophotometry. The reaction was followed at two wavelengths: 560 @
Abstract published in Advance ACS Abstracts, September 1, 1995.
0022-3654/95/2099-14365$09.00/0
nm to monitor Mn04- and 320 nm to monitor the accumulation and disappearance of reaction intermediates. The effects of acceleration and inhibition as a function of [H2C204]0 and [H2S04]0 can be visualized at 560 nm from t1/2(560), the time of half-disappearance of permanganate, and at 320 nm with t1,2(320) the time of half-disappearance of the intermediates. Figure 1 shows the geometrical representation of these two half-times. At the end of the reaction, the two curves are decaying but do not fall to zero at exactly the same time. Various amounts of intermediates remain when permanganate is used up. 1. Influence of [HzS04]0. An increase in the initial concentration of sulfuric acid leads to a parallel fall in t1/2(560) and t1/2(320)(Figure 2). 2. Influence of [HzCzO.&. The influence of the initial concentration of oxalic acid on the two values of t l / 2 is more complex, as it depends on the concentration domain under consideration: (a) At low concentrations when (1.25 x I [H2C204]0 5 1.6 x mobL-'), oxalic acid has an inhibitory effect (i): an increase in [H2C204]0 leads to a slowing of the reaction reflected by a increase in both t1/2(560)and t1/2(320). (b) At higher concentrations when ([H2C204]0 > 1.6 x mobL-'), the inhibitory effect (i) is only indicated by the disappearance of intermediates, and t1/2(320) increases. On the other hand, t1/2(560) falls, corresponding to an acceleration (a) of the disappearance of permanganate (Figure 3). 3. Bistability in a CSTR. When the reaction is carried out in a CSTR, the bistable behavior depends on the residence time.4 However, this phenomenon is only observed at oxalic acid concentrations close to the stoichiometric ratio. Under these conditions, the reaction is bistable and takes a place at residence times (z) ranging from 30 to 50 s. At higher [H~C204]0,the considerably slower reaction is monostable with residence times extending to 400 s (Figure 4).
111. Reaction Mechanism 1. Skeleton Mechanism. In a previous study,5 we demonstrated the presence of two different intermediates, Mn(1II) and 0 1995 American Chemical Society
Pimienta et al.
14366 J. Phys. Chem., Vol. 99, No. 39, 1995 m,
1.5
--
-
I
v) 400
1.0
X'
400
ili
9 0.5
0.0
0
Mx)
600,
I
Time (SI
Figure 1. Representation of t l ~ ( 3 2 0 (a) ) and t1/2(560)(b) on the plots of absorbance at 320 and 560 nm for the permanganate/oxalic acid reaction in sulfuric acid. (- - -) plot of absorbance of intermediates after subtracting absorbance of permanganate.
X
v -2
-1
0
O -3
log (H2C2041, Figure 3. (A) Plot of r1/2(320) against 1og[H2C204]0 for the permanganate/oxalic acid reaction in sulfuric acid [Mn04-]o = 5 x mol.L-', [H2S04] = 9 x low2mo1.L-I; Note the effect of continuous inhibition (i). (B) Plot of tIl2(560)under the same conditions. Note the inhibitory effect at low [H~C204]0(i) and the accelerating effect at high [ H ~ C Z O(a). ~ ] ~(0)Experimental points; ( x ) numerical simulation using model M7.
I
200
" 0.0
01
0.2
[H2S04l0
Figure 2. (A) Plot of t1/2(320)against [H2S04]0for the permanganate/ oxalic acid reaction in sulfuric acid; (0)experimental points; ( x ) numerical simulation using model M7. (B) Plot of t1/2(560)under the same conditions.
a soluble Mn(1V) species, whose exact structure could not be determined, but which was attributed to a variety of MnO2. We showed that this Mn(IV) species is reduced to Mn(1II) in a mixture of oxalic and sulfuric acids. Since we did not detect Mn(V1) and Mn(V) oxidation states, a skeleton scheme consisting of three overall steps was drawn up involving a total of four oxidation states of manganese: VI1
+ I1 - I11 + IV IV
-
111
(1)
(2)
111 I1 (3) We will describe our study of the mechanism of each of these steps: Steps 2 and 3 can be readily studied independently and then
Figure 4. Diagrams of stationary states of oxidation of oxalic acid by potassium permanganate in sulfuric acid, (Abs(560) stationary states as a function of residence time (t)in the CSTR. [Mn04-]0 = 5.6 x mol.L-', [H2S04]0 = 0.09 mo1.L-I. (A) [HzC204]0 = 1.5 mo1.L-' ( x stoichiometric ratio), bistability; (B) [H2C~04]0= 3.3 mo1.L-I (excess), monostability.
-
combined with step 1 to construct the overall reaction mechanism. The mechanism was analyzed in ascending order of the oxidation states of manganese: Mn(III), Mn(IV), and Mn(VII). Two kinetic studies reported in ref 5 enabled us to establish the mechanisms of reduction of the Mn(II1) intermediates (step 3) and Mn(IV) (step 2). 2. Model M3 of the Reduction of Mn(II1). To describe the reduction of Mn(II1) to Mn(II), our mechanism was based on the results of T a ~ b e and ~ . ~Noyes8 However, in order to simplify the description and restrict the number of steps, we only included observable species or those strictly required to describe the dynamics of the system and the stoichiometric equations. This mechanism involves the dissociation equilibria of the two diacids along with the equilibrium of complexation of Mn(1II):
J. Phys. Chem., Vol. 99, No. 39, 1995 14367
The K M ~ O ~ / H Z C ~ O ~ / H Reaction ZSO~
Figure 5. Comparison between tl/l(exp) and tl&im) in the kinetic study of the reduction of Mn(II1) into Mn(I1) by oxalic acid in sulfuric acid. The experimental conditions and simulations used are listed in mo1.L-I. The numbering of the experiments Table 2. [IIIz] cz 6 x is the same as that in ref 5.
The two complexes, monooxalatomanganese(II1) ([Mn(C~04)]+or III1) and bis(oxalatomanganate(III)) ([Mn(C204)2]or 1112) have differing reactivities: 1111 is reduced into Mn(I1) at a greater rate than is III2. The system will be then sensitive to the concentration of oxalate ions and thus to the concentrations of the two diacids, which in turn influence [ C Z O ~ ~ - ] depending on the positions of their dissociation equilibria. As [C204,-] rises, the complexation equilibria are displaced toward formation of I112 and the reduction reaction is slowed. In contrast, at low [ C ~ 0 4 ~ the - ] proportion of I112 falls and the reduction reaction speeds up. Sulfuric acid acts by displacing the dissociation equilibria of the two diacids: as [H2S04] rises, [C204,-] falls and the reduction speeds up. This point was discussed in more detail in ref 5 , but the what is that parameters for this reaction were selected and fitted numerically for three independent experimental kinetic studies using a set of common parameters. This produced an estimate of the kinetic parameters of model M3. In order to check the validity of this method using these parameters, we compared the results of nine kinetics studies at different [H2C204]0 and [HzS04]0. Since the kinetics were essentially of first order, they could be characterized by their values of tln. The corresponding values of tln(exp) have been compared with the values of tl/z(sim) using the model M3. It can be seen in Figure 5 that model M3 can accurately predict all values of t112 and is likely to incorporate the main reaction steps in the reduction of Mn(II1) into Mn(I1) by oxalic acid in sulfuric acid medium. 3. Model M4 for Reduction of Mn(1V). The second intermediate was a soluble variety of Mn02, whose exact structure was not determined. However, we showed that its reactivity toward mixture of oxalic and sulfuric acids was similar to that of MnO2 from the Guyard rea~tion.~ This compound formed a complexlo with molecular oxalic acid prior to its reduction into Mn(II1). The M4 mechanism is thus described by the two following processes: MnO, (2H')
+ H2C204
+ [Mn02,H,C,04]
-
f
[MnOz,H2C,0,]
Mn3+
+ CO, + (20,'- + 2HZO (1)
Apart from the two steps (H) and (I) mentioned above, mechanism M4 also involves the dissociation equilibria of the two diacids. Parameters were chosen for this model on the basis of three independent studies of the kinetics of the reduction of MnOz by oxalic acid in sulfuric acid. As in the M3 model study,
0.19 0.06 0.19 0.02 0.06 0.19
0.12 0.06 0.19
3
3.3 10-5 2.2 x 10-5 2.4 x 10-5
0.022 0.022 0.045
0.022
1.0 10-5 1.3 x 10-5 1.5 10-5 1.1 10-5 6.9 x 7.5 x 10-6
0.045 0.09 0.09 0.09 0.18
t112exp (SI
13
Figure 6. Comparison between tl/z(exp) and t&m) in the kinetic study of the reduction of Mn(1V) in oxalic/sulfuric acid mixtures. The experimental conditions and simulations used are listed in Table 3. [Mn02]0 = 0.6 x lo-' mo1.L-I. The numbering of the experiments is the same as that in ref 5. no.
[HzCz041o
j k
3 x 10-4 2.3 x 10-4 3 x 10-4 4.5 x 10-4 3 10-4 1.4 10-4 2 x 10-4 3 x 10-4 5 10-4 2 x 10-4 3 x 10-4
[HzS04lo 0.09 0.09 0.045 0.022 0.022 0.045 0.022 0.01 1 0.006 0.01 1
0.006
W2C2041eq 2 x 10-4 1.5 10-4 1.5 x 10-4 1.5 10-4 1.1 10-4 7.1 x 10-5 7.1 x 10-5 7.1 x 10-5 7.1 x 10-5 4.4 4.4
10-5 10-5
since the kinetics were essentially of first order, they could be characterized by their values of t i l 2 exp (Figure 6 ) . It can be seen that model M4 can accurately predict all values of t l and ~ is therefore likely to incorporate the main reaction steps in the reduction of Mn(IV) into Mn(II1) by oxalic acid in sulfuric acid medium. 4. Complete Mechanism for the PermanganatdOxalic Acid Reaction in Sulfuric Acid Medium (M7). To establish a complete model of the permanganate/oxalic acid reaction in sulfuric acid, the kinetic schemes M3 and M4 were combined with that for the reduction of permanganate. With respect to
Pimienta et al.
14368 J. Phys. Chem., Vol. 99, No. 39, 1995
uj
9 \
\
\ \
0
0
400
XK)
600
Time (s)
Figure 7. Absorbance at 560 nm during reduction of MnO4- by oxalic mol.L-', [H2S04]0= 0, acid in sulfuric acid. [Mn04-]0 = 5 x 27 mol.L-', [H2C204]0 = 1.25 x mo1.L-I. ( 0 )Experimental points; (- - -) fitting: (1) without step (G); (-) fitting: (2) with step (G).
the latter process, the only chemically realistic mechanism is that described by Adler et al. (ref 8), which can be summarized by the following processes E and F:
Mn2+
(4H')
+ C,0,2-
MnC,O,
(E)
+ Mn0,- + MnC20, -MnO, + Mn3+ + 2C02 +
2H,O (F) Process E corresponds to formation of the monooxalatomanganese(II) complex," MnC204, while process F is the reduction of permanganate by the complexed Mn(I1) leading to formation of Mn(II1) and Mn02. To simplify the expression of the rate of process (F), we assumed that the rate-limiting step was the interaction of permanganate ions with complex MnC204. However, for conservation of matter, the consumption of four protons must be included each time an Mn04- ion reacts with an MnC204 complex. We initially studied a model constructed by combining schemes M3 and M4 and processes E and F. However, this did not completely reproduce the kinetics of permanganate even after refitting all the values of the parameters [Figure 7). This indicated that further steps were required in the model. The simplest expedient was to add a direct reduction of permanganate by molecular oxalic acidI2 according to the following equation: (4H')
+ Mn04- + (2)H2C204- Mn3+ i-4 c 0 , + 4H2O
(GI During this process, which is the sum of several elementary steps, four protons and two molecules of H2C2O4 are consumed. We assumed that the rate-limiting step was the interaction between Mn04- ions and molecular oxalic acid. The rate of this step (G) is given by the relationship:
Figure 7 shows that the kinetics of disappearance of Mn04was accurately reproduced by taking process G into account. Nevertheless, the values of the parameters applicable over the whole experimental domain cannot be extracted from a fit of parameters from a single experiment. On the basis of these observations, we attempted to model the mechanism of the complete permanganate/oxalic acid
sulfuric acid reaction according to the following scheme, referred to hereafter as scheme M7 (Figure 8). 5. Estimation and Fitting of Parameters for Scheme M7. In order to estimate the parameters of model M7 and generate acceptable predictions from the model, two independent reactions were carried out for different values of [H2C204]0. A concentration zone was selected in which the inhibitory effect of oxalic acid was marked. The two reactions were followed at 320 and 560 nm in order to assay the kinetics of both permanganate and intermediates. Some of the parameters in the model M7 set had already been estimated from models M3 and M4, namely, the values of KH and k1 (M4) and Kj, k ~ k, ~ k, ~ and , k~ (M3) the equilibrium constants for dissociation of the two diacids KA, KB, Kc, and KD, were obtained from published values of pKa. The molar extinction coefficients of Mn04- and 1112 were estimated from independent measurements = 1400 Lmol-km-', = 1660 L-mol-km-', ;::2 = 0 Lmol-km-', c;? =" 2500 Lmol-km-'. Those of Mn(IV), were estimated {rom the spectral decompositions reported in ref 5: E 560~ ~ =( ~0 ~Lmol-'*cm-', ) E$&) = 2500 Lmol-km-' , these two latter values were employed for both free Mn(IV), and for the [Mn(IV),, H2C2041 complex. The remaining parameters KE, k ~ and , kc were fitted numerically from the two reactions illustrated in Figure 9 using differential equations shown in Table 1.
$to
EE:~
TABLE 1: Expressions for the Reaction Rates of the Steps in Mechanism M7 along with the Kinetic Equations for the Corresponding Species step
rate
observation
The KMn04/H2C204/H2SO4Reaction
J. Phys. Chem., Vol. 99, No. 39, 1995 14369 K,,= 5 10” mo1.P KB= 5 IO” mo1.T’ &=n
KD= 1.27 10” mo1.T’
KE= 1.6 lO+’mol-’.l
I
(4g)+ Mn04- + (2) H2C204 + Md++ 4 COz + 4 H20
I
kF = 5
mol-’.l.s-’
=2
l o 2 moP.l..d
KH=1.77 loMmorI.1 kI = 8 lo-’ s-’
KJ= 2
mol-l.1
KK= 1.29 10’’ mol-l.1 kL = 0.92 s-’ kM = 1.3
kN =
10” s-l mol-I.1.s-l
Figure 8. Scheme M7 for reduction of permanganate by oxalic acid in sulfuric acid. Frames (E) to (G), (H) and (I), and (J) to (N) correspond to the overall processes 1, 2, and 3, respectively, of the skeleton mechanism. (*): strong acid. 1s I
I
0
a 320
0
0
LO
-
0
b 320
0
0
0
0
0
0 0
0 0
0
, * e 8 ~ o o o o o o
vj
a 4
0
’ ~ e ~ o o o o o 0
0.5 .
O O O
0 0
0
O O O 0 0
0
a560
0
o
0
b 560
IV. Predictive Power of Model M7
0 0
0 0
0
0
0
0
0 0
0 .
0.0
vj
n 4
0.5
0
Time
To validate model M7 with these parameters, we examined
A*a its behavior over the whole experimental domain.
LO
0.0
species and the inhibitory effect of oxalic acid. However, the rigor of this multiexperiment fitting reveals its imperfections. They may be attributed to experimental error, improper definitions or neglect of additional processes such as those involving other oxidation states of manganese. Nonetheless, we showed that within the selected experimental domain, scheme M7 using these parameters predicted the halfreaction times t1/2(320) and t1/2(560), as well as the monostable or bistable behavior in the CSTR as a function of [H2C204]0.
(SI
Figure 9. (A) Absorbance at 560 and 320 nm during the reduction of permanganate by oxalic acid in sulfuric acid. [Mn04-]0 = 4.5 x mol.L-’, [H2S04]0 = 0.09 x mo1.L-I. [H2C204]0 = (a) 1.5 x mol.L-’, (b) 6.2 x mo1.L-I. (B)Corresponding simulations in model M7 using the parameters of Figure 8. [MnO& = 4 x mo1.L-I.
It can be seen that model M7 reproduces the overall kinetics of the two reactions, namely the accumulation of intermediate
1. Half-Reaction Times t1/2(320) and t1/2(560). Figure 2 shows that the accelerating effect of [H2S04]0 was predicted with respect to both the disappearance of intermediates (tln(320)) and permanganate (t1/2(560)). As [H2S04]0 increases, the displacement of the equilibria A, B, C, and D leads essentially to a fall in concentration of oxalate ions and thus a speeding up of the reduction of Mn(II1) into Mn(II). The absorbance at 320 nm, due mainly to 1112, falls at an increasing rate: t1/2(320) falls as [H2S04]0 increases. This acceleration of the reduction of Mn(II1) into Mn(I1) also leads to an enhancement of the autocatalflc effect by the intermediate of the cycle of reactions J, K, L, M, E, and F: permanganate decays more rapidly and t1/2(560) falls. Figure 3 shows that the effect of [H2C204]0 is also predicted: it depends on the concentration domain under consideration and the observation wavelength. At low oxalic acid concentration, as [H2C204]0 rises [C2042-] also rises and the disappearance of Mn(III) is slowed: f1/2(320) rises. Under these conditions, production of Mn(II) is inhibited, autocatalysis is attenuated, and t1/2(560) also rises. When the oxalic acid concentration is elevated, two opposing processes occur together: (a) a residual inhibitory effect on the reduction of Mn(II1): t1/2(320) rises; (b) the rate of direct reduction of
14370 J. Phys. Chem., Vol. 99, No. 39, 1995
Pimienta et al. Nevertheless, it should be bome in mind that there is a certain degree of uncertainty in the individual values of the kinetic parameters. This is indicated by a possible covariance between two or more parameters. However, the values proposed here are chemically realistic and are in line with the available literature data on this reaction. Various elements of this reaction scheme could perhaps be employed to elucidate the mechanism of the oscillators based on the chemistry of manganese, which have been described re~ent1y.I~
Figure 10. Simulationusing model M7 and parameters of Figure 8 of the diagram of the stationary states of the oxidation reaction of oxalic acid by potassium permanganate in sulfuric acid (Abs(560) as a function of residence time (t)in the CSTR). The initial concentrations are the same as those in Figure 4.
permanganate by molecular oxalic acid (step G) increases: t ~ ( 5 6 0 falls. ) 2. Bistability in a CSTR. The reaction exhibits bistable behavior at low [H2C204]0 and monostable behavior at high [H2C204]0. Model M7 can reproduce this nonlinear behavior, and it predicts the appearance of bistability in a CSTR at low oxalic acid concentrations and loss of bistability on increase in concentration of this acid (Figure 10). This phenomenon can be accounted for by the autocatalytic loop incorporated in the skeleton mechanism along with the inhibitory steps involved in the reduction of Mn(II1). When the initial concentration of oxalic acid is close to the stoichiometric ratio (1/2.5), its concentration drops progressively to zero. This removes the inhibitory influence on the reduction of Mn(III) and there is an increase in the rate of production of Mn(I1). This phenomenon cooperates with autocatalysis and produces a sufficient acceleration at the end of the reaction to give rise to bistability in a CSTR. On the other hand, if the initial concentration of oxalic acid is high, the relative change in concentration of oxalate ions during the reaction is negligible, the inhibitory effect persists, and the reaction remains monostable as there is insufficient accelerating action from the autocatalytic loop alone.
V. Conclusion An overall mechanism for the reduction of permanganate by oxalic acid in sulfuric acid medium involving the following four main processes was drawn up: (1) the dissociation equilibria of the two diacids, sulfuric acid and oxalic acid, in aqueous solution, (2) the steps in the reduction of Mn04- ions either by undissociated oxalic acid leading to the formation of Mn(II1) or by complexed Mn(II) leading to the formation of both Mn(III) and Mn(IV), (3) the reduction of Mn(1V) into Mn(II1) after combination with undissociated oxalic acid, and (4) the complexation equilibria of Mn(1II) by oxalate ions and the reduction of the corresponding complexes into Mn(I1). The rate of this reduction is a function of the degree of complexation: the higher the degree of complexation, the slower the reaction. Within the domain under consideration, model M7 using the parameters of these steps was able to predict the half-reaction times as a function of the initial concentrations of sulfuric and oxalic acids. The acceleration and decay of MnO4- induced by concentrated oxalic acid is the result of a direct action of molecular oxalic acid on permanganate ions. This mechanism could also account for the bistable behavior observed in a CSTR,I3 stemming from a cooperative effect between autocatalysis and removal of the inhibitory influence on the reduction of Mn(1II) by oxalate ions.
VI. Experimental Part
1. Preparation of Solutions. All the solutions were prepared from reagents of the highest available purity. KMn04 (M = 158,03); MnS04,H20 (M = 169,02); Mn(CH3C00)3,2H20 (M = 268,1), H2S04 (M= 98,07); H2C204,2H20 (M = 126,07) were dissolved in double-distilled water. The solution of permanganate was acidified with [HzSO~]= 0.0225 m0l.L-l to inhibit formation of Mn02 precipitate. 2. Spectrophotometric Recording of Reaction Kinetics. Measurements were carried out at 25 "C on 2 mL samples in a 1 cm x 1 cm quartz cuvette. Spectra were recorded in a diodearray spectrophotometer (HP 8451) connected to a HP 9000/ 380 or 710 series workstation running UNIX. 3. CSTR. ( a ) Pumps. Two identical HPLC pumps, one for oxalic acid and sulfuric acid, the other for permanganate, with flow rates adjustable over a range of 0.01 to 5 mL min-' were used. (b)Reactor. The reactor was a quartz cylinder closed at either end with Teflon bungs. The lower stopper was fitted with two injectors, and the upper contained an inverted cone to evacuate bubbles. The contents were stirred continuously with a magnetic bar driven by a stepper motor whose speed could be controlled. The internal volume of the reaction was 2.1 mL. The whole setup was enclosed in a thermostated copper block placed inside the sample chamber of the spectrophotometer. The reactor had an optical path length of 1.1 cm. VII. Data Processing and Analysis
1. tyz(exp). tln(exp) in the experimental study of the kinetics of reduction of Mn(1II) into Mn(I1) and of MnO2 into Mn(1II) by oxalic acid in sulfuric acid medium was determined by fitting the kinetics from the following first order equation: Abs = (Abs,,,
+
- Abs,) e-kobsf Abs,
t ~ d e x p )= In 2kbs. 2. Simulation and Fitting. To simulate the models, the differential equations governing the kinetics of all the proposed chemical species were integrated numerically (semiimplicit Runge-Kutta method).I5 Each of the fast dissociation or complexation equilibria was split into its two opposite steps, the ratio of the rate constants giving the corresponding equilibrium constant. Table 1 lists details of these equations. Using Beer-Lambert's law we calculated the change in absorbance from the concentrations and molar extinction coefficients of the absorbing species. The parameters of the model comprised the rate constants and initial concentrations of the species in solution. Some parameters were already known and the rest were fitted numerically using an iterative algorithm of the Powell type.I6 The residual error (RE) = CjXp(k&c) q(&s))*/jp,was minimized wherej is the number of points for each kinetic determination and p the number of kinetics fitted simultaneously.
The K ~ ~ - I O ~ / H ~ C ~ O Reaction &I~SO~
To simulate the behavior of the reaction in a CSTR, a flux term was introduced into each of the kinetic equations in the model. The flux term is always of form: %(Xi0 - Xi) where & = l/z in the CSTR. Xio is the concentration of compound i at the input and Xi its concentration inside and at the output of the reactor. References and Notes (1) Harcourt, A. V.; Esson W. Philos. Trans. R. SOC.London 1866, 156-93. (2) Reyckley, J. S.; Showalter, K. J . Am. Chem. SOC.1981,103,70123. (3) de Kepper, P.; Ouyang, Q.; Dulos, E. In Nonequilibrium Dynamics in Chemical Systems; Vidal, C., Pacault, A,, Eds.; Springer: New York, 1984; p 44-9. (4) Pimienta, V.; Lavabre, D.; Levy, G.; Micheau, J. C. J . Phys. Chem. 1992, 96, 9298-301. ( 5 ) Pimienta, V.; Lavabre, D.; Levy, G.; Micheau, J. C. J. Phys. Chem. 1994, 98, 13294-9. Note that letters k and m in table I and letters r and t in Figure 4b must be inverted. (6) Taube, H. J . Am. Chem. SOC.1947, 69, 1418-28. (7) Taube, H. J . Am. Chem. SOC.1948, 70, 1216-20. (8) Adler, S. J.; Noyes, R. M. J . Am. Chem. SOC.1955, 77, 2036-42. (9) Guyard, A. Bull. SOC.Chim. Fr. 1864, I , 89-93.
J. Phys. Chem., Vol. 99,No. 39, 1995 14371 (10) Bradley, J.; Van Praagh, G. J . Am. Chem. SOC.1938, 1624-36. (11) Money, R. W.; Davies, C. W. Trans. Faraday. SOC.1932,28,60914. (12) (a) Brillas, E.; Garrido, J. A,; Perez-Benito, J. F.; Rodriguez, R. M.; De Andres, J. Collect. Czech. Chem. Commun. 1988, 53, 479-486. (b) Rodriguez, R. M.; de Andres, J.; Brillas, E.; Garrido, J. A,; Perez de Benito, J. New J. Chem. 1988, 12, 143-46. (13) Gray, P.; Scott, S. K. Chemical Oscillations and Instabilities, Non Linear Chemical Kinetics; Clarendon Press: Oxford, U.K., 1990; ISBN: 0- 19-855646-2. (14) (a) Nagy, A.; Treindl, L. Nature 1986,320,344-5. (b) de Kepper, P.; Ouyang, Q. Chemical Reaction in Liquids; Moreau, M., Turq, P., Eds.; Plenum Press: New York, 1988; pp 459-67. (c) Morita, M.; Iwamoto, K.; Seno, M. Bull. Chem. SOC.Jpn. 1988, 61, 3467-70. (d) Nagy, A.; Treindl, L. J. Phys. Chem. 1989, 93, 2807-10. (e) Orban, M.; Epstein, I. R. J . A m . Chem. SOC.1990, 112, 1812-17. ( f ) Doona, C. J.; Kustin, K.; Orban, M.; Epstein, I. R. J . Am. Chem. SOC.1991, 113, 7484-9. (g) Melichercik, M.; Mrakavova, M.; Nagy, A.; Olexova, A.; Treindl, L. J . Phys. Chem. 1992, 96, 8367-68. (h) Fazekas, T.; Nagy, A,; Treindl, L. Collect. Czech. Chem. Commun. 1993, 58, 775-82. (15) (a) Kaps, P.; Rentorp, P. Numer. Math. 1979, 33, 55-68. (b) Gottwald, B. A. Computing 1981, 26, 355-60. (c) Kaps, P.; Rentorp, P. Comput. Chem. Eng. 1984, 8, 393-6. (16) Minoux, M. Programmation mathtmatique; Vol. 1, Chapter 4, Dunod: Paris, 1983; ISBN: 2-04-0.15487-6.
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