Malonic acid concentration as a control parameter in the kinetic analysis of the Belousov-Zhabotinsky reaction under batch conditions

Article abstract of DOI:10.1039/b804919j

The influence of the initial malonic acid concentration [MA]₀ (8.00 × 10^-3≤ [MA]₀≤ 4.30 × 10 ^-2 mol dm^-3) in the presence of bromate (6.20 × 10^-2 mol dm^-3), bromide (1.50 × 10^-5 mol dm^-3), sulfuric acid (1.00 mol dm^-3) and cerium sulfate (2.50 × 10^-3 mol dm^-3) on the dynamics and the kinetics of the Belousov-Zhabotinsky (BZ) reactions was examined under batch conditions at 30.0°C. The kinetics of the BZ reaction was analyzed by the earlier proposed method convenient for the examinations of the oscillatory reactions. In the defined region of parameters where oscillograms with only large-amplitude relaxation oscillations appeared, the pseudo-first order of the overall malonic acid decomposition with a corresponding rate constant of 2.14 × 10^-2 min^-1 was established. The numerical results on the dynamics and kinetics of the BZ reaction, carried out by the known skeleton model including the Br₂O species, were in good agreement with the experimental ones. The already found saddle node infinite period (SNIPER) bifurcation point in transition from a stable quasi-steady state to periodic orbits and vice versa is confirmed by both experimental and numerical investigations of the system under consideration. Namely, the large-amplitude relaxation oscillations with increasing periods between oscillations in approaching the bifurcation points at the beginning and the end of the oscillatory domain, together with excitability of the stable quasi-steady states in their vicinity are obtained. the Owner Societies.

Full text of DOI:10.1039/b804919j

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Malonic acid concentration as a control parameter in the kinetic analysis
of the Belousov–Zhabotinsky reaction under batch conditions
a
b
c
a
ˇ
ˇ
, Zeljko D. Cupic
Slavica M. Blagojevic
´
and Ljiljana Z. Kolar-Anic
,* Slobodan R. Anic
´
b
´
, Natas
ˇ
a D. Pejic
´
´
Received 25th March 2008, Accepted 22nd August 2008
First published as an Advance Article on the web 30th September 2008
DOI: 10.1039/b804919j
ꢁ
3
The influence of the initial malonic acid concentration [MA]
0
(8.00 ꢀ 10 r [MA]
0
r
.30 ꢀ 10ꢁ mol dm ) in the presence of bromate (6.20 ꢀ 10 mol dm ), bromide
2
ꢁ3
ꢁ2
ꢁ3
4
ꢁ5
ꢁ3
ꢁ3
ꢁ3
ꢁ3
(
1.50 ꢀ 10 mol dm ), sulfuric acid (1.00 mol dm ) and cerium sulfate (2.50 ꢀ 10 mol dm
)
on the dynamics and the kinetics of the Belousov–Zhabotinsky (BZ) reactions was examined under
batch conditions at 30.0 1C. The kinetics of the BZ reaction was analyzed by the earlier proposed
method convenient for the examinations of the oscillatory reactions. In the defined region of
parameters where oscillograms with only large-amplitude relaxation oscillations appeared, the
pseudo-first order of the overall malonic acid decomposition with a corresponding rate constant of
2
ꢁ1
2
.14 ꢀ 10ꢁ min was established. The numerical results on the dynamics and kinetics of the BZ
2
reaction, carried out by the known skeleton model including the Br O species, were in good
agreement with the experimental ones. The already found saddle node infinite period (SNIPER)
bifurcation point in transition from a stable quasi-steady state to periodic orbits and vice versa is
confirmed by both experimental and numerical investigations of the system under consideration.
Namely, the large-amplitude relaxation oscillations with increasing periods between oscillations in
approaching the bifurcation points at the beginning and the end of the oscillatory domain, together
with excitability of the stable quasi-steady states in their vicinity are obtained.
1
. Introduction
previously proposed procedure successfully applied on the
The obtained kinetic results enable
25,26
Bray–Liebhafsky system.
1
,2
The Belousov–Zhabotinsky (BZ) reaction,
namely the
further investigation of the BZ oscillator where malonic acid
concentration is considered as the control parameter defining the
dynamic states and bifurcation point at the beginning and at the
end of the oscillatory region. The character of both bifurcation
points is examined by periods, oscillatory amplitudes and excit-
ability of the stable quasi-steady states in their vicinity, as already
performed in previous investigations of the Belousov–Zhabotinsky
decomposition of organic acid, by bromate ions in acid solutions
containing metallic catalysts has attracted both experimental
3–15
and theoretical interest for a long time.
In fact, this is the
most investigated oscillatory chemical reaction in a closed and
open reactor, the reaction where the complex dynamic states
including oscillatory, mixed mode and chaotic ones are found in
the narrow region of reactant concentrations. Therefore, the
influence of reactant concentrations, particularly organic sub-
strate undergoing decomposition, is analyzed from the early
27,28
system in a close reactor performed by Noszticzius et al.
Furthermore, all our experimentally obtained results are
simulated numerically. For this purpose, the slight modifica-
tion of the existing models was necessary. In the variant of the
model applied here, instead of either the direct autocatalytic
16–24
discovery of the oscillatory phenomena in this system.
Here we focus on the kinetics of malonic acid decomposition
and the influence of the initial malonic acid concentration on the
overall process in the presence of bromate, bromide and cerium
ions in the well stirred close reactor at a temperature of 30.0 1C.
The formal kinetics of the overall BZ reaction is examined by the
step or the variable stoichiometric coefficient f present in some
29
models, the Br O appeared as intermediate species.
2
2
. Experimental
a
Faculty of Pharmacy, Department of Physical Chemistry, University
The kinetics of the isothermal BZ oscillatory reaction (30.0 1C)
was studied in thermostated batch reactor (Metrohm
of Belgrade, Vojvode Stepe 450, Belgrade, 11000, Serbia
E-mail: slavica.blagojevic@pharmacy.bg.ac.yu.
E-mail: nata@pharmacy.bg.ac.yu; Fax: +99 381 11 3972 840;
Tel: +99 381 11 3951 286
a
ꢁ3
3
EA–876–20) with a reaction volume of 51 ꢀ 10 dm equipped
with a magnetic stirrer. The stirring rate was kept at 700 rpm. A
bromide ion-sensitive electrode (Metrohm EA 306) versus a
double junction Ag/AgCl electrode (Metrohm model 6.0726.100)
as the reference one was used for potentiometric monitoring of the
time evolution of the BZ system (MA 6370 potentiometer, Iskra,
Slovenia). The temperature was controlled within ꢂ0.1 1C by a
circulating water thermostat (Series U, MLW Freital, Germany).
b
Faculty of Physical Chemistry, University of Belgrade, P.O. Box 47,
Belgrade, 11000, Serbia. E-mail: lkolar@ffh.bg.ac.yu.
E-mail: boban@ffh.bg.ac.yu; Fax: +99 381 11 2187 133;
Tel: +99 381 11 3336 876
c
IChTM, University of Belgrade–Department of Catalysis and
Chemical Engineering, Njegosˇeva 12, Belgrade, 11000, Serbia.
E-mail: zcupic@msn.com; Fax: +99 381 11 2637 977;
Tel: +99 381 11 2630 213
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Analytical grade reagents and deionized water with a specific
resistance of 18 MO cm (Milli-Q, Millipore, USA) were used for
preparing the solutions.
period characterized by the large-amplitude relaxation oscilla-
tions appearing between t
1
and tend where tend is the time
interval between the beginning of the reaction and the end of
The experimental conditions are selected such that oxygen
the last incomplete oscillation (tend = tend ꢁ t = tend
)
0
3
0
inhibition is negligible. Namely, the oxygen influence on the
reaction system is examined. After performing parallel experi-
ments with purification of the system with a stream of nitrogen
through the solution before the start of the reaction and
without it, we did not find significant difference, and, there-
fore, continue the experimental investigations without it.
The experiments were carried out with various initial malonic
(Fig. 1(c)). The mentioned periods, characteristic of oscillatory
reactions are direct consequence of variable malonic acid
concentration due to its decomposition. Under the considered
conditions we have never observed damped oscillations at the
end of the oscillatory domain that should be inappropriate for
the determination of tend. The typical potentiometric traces of
the investigated BZ systems at different initial concentrations
ꢁ
3
acid concentrations (8.00 ꢀ 10
r [MA]
mol dm ). The initial concentrations of potassium bro-
mate, sulfuric acid, potassium bromide and cerium sulfate were
0
r 4.30 ꢀ
of malonic acid ([MA]
Obviously, in the interval of [MA]
0
) are illustrated in Fig. 1.
ꢁ
2
ꢁ3
ꢁ3
1
0
0
between 9.00 ꢀ 10 and
ꢁ
2
ꢁ3
1.20 ꢀ 10 mol dm , there is the critical initial malonic acid
concentration ([MA]0,crit) corresponding to the transition
between the non-oscillatory and oscillatory evolution of malo-
nic acid decomposition; and from this point of view, denoting
ꢁ
3
kept fixed in all experiments: [H SO ] = 1.00 mol dm
2
,
=
4 0
ꢁ2
ꢁ3
[
KBrO ]
3
=
6.20
ꢀ
10
mol dm ,
[KBr]0
0
5
ꢁ3
ꢁ3
ꢁ3
1
.50 ꢀ 10ꢁ mol dm , [Ce (SO ) ] = 2.50 ꢀ 10 mol dm
.
3
2
4 3 0
ꢁ
3
The BZ reaction was started after injection of 1 ꢀ 10 dm
some kind of the bifurcation point similar to, but not the same
27,28
ꢁ
3
3
of cerium sulfate solution into the 50 ꢀ 10 dm of the
mixture containing malonic acid, potassium bromate, sulfuric
acid, and potassium bromide solutions of the concentrations
mentioned at the moment when the potential of the bromide
sensitive electrode achieved the steady value of about 200 mV.
as, the one obtained in the open reactor.
experiments carried out with [MA]
Namely, the
0
4 [MA]0,crit possess
the oscillatory region. Thus, under the conditions considered,
we can infer that, with the increase of the malonic acid
concentrations, the time evolution of the BZ system in the
0
point [MA] = [MA]0,crit shifts from the one corresponding to
the stable steady state to the one corresponding to the limit
cycle oscillations or vice versa. The number of observed
oscillations increases with further increase of the initial malo-
nic acid concentration. We noticed that their form, indepen-
dently of their number, corresponds to the large-amplitude
relaxation oscillations. Moreover, if all the other reagent
3
. Results and discussion
3
.1 Experimental results
The evolution of the BZ oscillator takes place through one
preoscillatory period (t = t = t and one oscillatory
ꢁ t
1
1
0
1
Fig. 1 Under the heading Experimental results are presented the potentiometric traces of the bromide ion-sensitive electrode, in the order of
ꢁ2
ꢁ
3
ꢁ3
ꢁ3
ꢁ3
ꢁ2
ꢁ2
ꢁ2
increasing [CH
2
2
(COOH)
2
]
0
(in mol dm ): (a) 8.00 ꢀ 10 , (b) 9.00 ꢀ 10 , (c) 1.20 ꢀ 10 , (d) 1.60 ꢀ 10 , (e) 2.20 ꢀ 10 , (f) 3.20 ꢀ 10
,
ꢁ
ꢁ2
ꢁ3
ꢁ5
ꢁ3
ꢁ3
ꢁ3
(
g) 4.30 ꢀ 10 . [KBrO
3
]
0
= 6.20 ꢀ 10 mol dm , [H
2
SO
4
]
0
= 1.00 mol dm , [KBr]
0
= 1.50 ꢀ 10 mol dm , [Ce
2
(SO
4
)
3
]
0
= 2.50 ꢀ 10 mol dm
ꢁ
at 30.0 1C. Under the heading Calculated results are presented the time dependences of log[Br ] obtained by the numerical simulation of the kinetic
scheme given in Table 1 in order of increasing initial concentration of malonic acid (in mol dm ) equal to the values noted in the Experimental
ꢁ3
+
0 0
] = 1.29 mol dm . The concentrations of the other external species are as in the experimental results. [HOBr] =
.50 ꢀ 10ꢁ mol dm , whereas the initial concentrations of other intermediates are zero in numerical simulations.
ꢁ3
results from (a) to (g). [H
8
ꢁ3
1
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ꢁ
3
ꢁ3
concentrations are kept constant, the [MA]0,crit would be in
some relation with [MA]end, the malonic acid concentration at
the end of any oscillogram (after the period tend elapsed).
The same type of potentiometric traces is an indication that
the kinetics of the BZ reaction in all cases under the considered
conditions is unchanged. Hence, in accordance with our
previous examinations of the Belousov–Zhabotinsky reac-
and the value for [MA]end is 6.02 ꢀ 10 mol dm [Fig. 2].
Obviously, relation (2) is valid for lower values of the initial
malonic acid concentrations {smaller than or equal to
ꢁ2
ꢁ3
ꢁ2
3.20 ꢀ 10 mol dm [ln(3.20 ꢀ 10 ) = ꢁ3.44]} indicating
that the proposed first order kinetics for malonic acid decom-
position with respect to itself is fulfilled in this region of [MA]0.
The deviation from the straight line shows that the kinetics of the
3
1
tions by the procedure proposed during the analysis of the
2
0
overall reaction changes for higher values of [MA] .
5,26,32,33
Bray–Liebhafsky reaction,
[
we could assume that
The value of the rate constant k suggests that the malonic
acid decomposition is changing slowly enough so that the time
scales of the oscillations are much shorter than the time scale
of relaxation of the overall reaction system in the batch
reactor. The obtained value for [MA]end denotes the threshold
between the oscillatory and non-oscillatory domains, that is,
the concentration of malonic acid in the bifurcation point,
since for any concentration higher than the mentioned one, the
main quasi-steady state will be unstable and some limit cycle
will appear. This malonic acid concentration must be in some
relation with [MA]0,crit, that corresponds to the interception
MA]end, the malonic acid concentration at the tend is practi-
cally constant, and independent of the initial malonic acid
concentration. Since all other reagent concentrations are
approximately constant, increasing of the initial malonic acid
concentration only prolongs the time elapsed from the begin-
ning of the reaction to the end of the last incomplete oscilla-
tion, that is, the tend. Having this in mind, and the fact that
reactions with reactants that undergo decomposition are often
first order with respect to them, we chose to analyze the
kinetics of the overall BZ reaction as the pseudo-first order
reaction with respect to malonic acid
between curves denoting tend and t
Fig. 2).
Having the rate constant of the overall process we were able
1 0
as function of [MA]
(
d½MAꢄ
¼
ꢁk½MAꢄ
ð1Þ
dt
to calculate the malonic acid concentrations in any instant,
including t . The values in the moment t depend on the initial
The influence of the other species on the overall reaction is
included in the complex rate constant k. In the integrated
form, the eqn (1) reads
1
1
malonic acid concentration. Since the malonic acid concentra-
tion is not constant in t , eqn (2) cannot produce the linear
1
relation between t and ln[MA] if t is replaced by t , and
[MA]end by [MA] (Fig. 2). However, the functions given in
1
1
0
end
1
1
1
tend ¼ ꢁ ln½MAꢄend
þ
ln½MAꢄ0;
ð2Þ
k
k
Fig. 2 will help us find the critical initial malonic acid
concentration necessary to provoke the oscillatory state in
the considered system. It is in the intersection of the curves t1
where [MA]0 and [MA]end denote the concentrations of
malonic acid in t = 0 and t = tend. From the experiments in
which only concentrations [MA] were varied, we easily obtained
the rate constant k of the overall BZ reaction from the straight
line shown in Fig. 2. The rate constant k is 2.14 ꢀ 10ꢁ minꢁ1
0
= f(ln[MA] ) and t
0
end 0
= g([lnMA] ), which presents the
critical point of the considered system under batch conditions
with respect to initial malonic acid concentration. In our case
2
,
ꢁ
3
ꢁ3
it is about [MA]0,crit = 9.20 ꢀ 10 mol dm . For [MA]
0
o
MA]0,crit, the oscillatory region is not found, whereas it exists
[
in the opposite case. Thus, Fig. 2 presents one kind of
bifurcation diagram for the dynamic system under batch
conditions where the initial malonic acid concentration is the
control parameter. On the other hand, the [MA]end is also
the concentration of malonic acid in the bifurcation point. If
we want to find a relation between [MA]0,crit and [MA]end, we
must take care about the fact that we are dealing in our
experiments with catalytic malonic acid decomposition under
batch conditions and that, therefore, the malonic acid
concentration must decrease with time. Hence, we need to
calculate the critical malonic acid concentration [MA]crit, just
in the bifurcation point, that is, in the point where t = t
=
1
end
tcrit, and, consequently, [MA] = [MA] = [MA]crit. Having
1
end
tcrit and rate constant of the overall process k, we can
easily calculate the [MA]crit from [MA]0,crit. It is equal to
ꢁ
3
ꢁ3
6
.12 ꢀ 10 mol dm . Hence, by the here applied analysis
Fig. 2 The time interval between the beginning of the reaction and
the end of an oscillatory evolution tend, and preoscillatory period t
obtained as functions of ln[MA] . The full circles and squares denote
the experimental results, whereas the empty symbols correspond to the
numerically simulated values. The interceptions of curves t
f(ln[MA] ) and tend = g([lnMA]
of the dynamic states in a where only transient dynamic states
are possible, we defined three mutually correlated values for
critical malonic acid concentrations: [MA]0,crit, [MA]crit and
1
0
[
MA]end, where the two last ones are equal and correspond to
1
=
ꢁ3
the bifurcation point.
0
0
) are at [MA]0,crit = 9.20 ꢀ 10
ꢁ3
ꢁ3 ꢁ3
mol dm and [MA]0,crit = 9.26 ꢀ 10 mol dm , for experimental
Thus, by experimental investigations of the influence of
initial malonic acid concentration on the dynamic states
data and numerical simulations, respectively.
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during time evolution of the Belousov–Zhabotinsky reaction,
we found some kind of the bifurcation points that denote the
thresholds between stable and unstable steady states under
batch conditions. (In the closed reaction systems we can speak
only of quasi-steady states.) Now we should like to know a
little about the type of bifurcation points under consideration.
This problem was already elaborated in the bromate–
cerium–oxalic acid oscillatory system, that is the other type
of Belousov–Zhabotinsky reaction under batch conditions, in
2
7
the paper by Noszticzius et al. and in the malonic acid BZ
system in ref. 28. They found that both bifurcations, in t and
1
in tend, are the saddle-node infinite period (SNIPER) ones,
what we confirmed here. In the vicinity of the bifurcation point
at the end of the oscillatory domain, only large-amplitude
relaxation oscillations (Fig. 1) with an increasing period
between them (Fig. 3) are noted if [MA] 4 [MA]end, whereas
when [MA] o [MA] the excitability is found (Fig. 4).
end
At the beginning of the oscillatory domain, the behavior of
the system is analogous but in reverse sequence. Thus, in the
preoscillatory period with [MA] 4 [MA]
1
the excitability is
found, whereas when [MA] o [MA] only large-amplitude
1
relaxation oscillations with increasing periods are noted as the
system approaches the bifurcation point. As we don’t have
small amplitude oscillations, we can say that we are dealing
with the SNIPER bifurcation points, as it is already obtained
under batch conditions in ref. 27 and 28. Consequently, the
same conclusion is valid for the transition between the non-
oscillatory time evolutions and the oscillatory ones with
respect to initial malonic acid concentration. An analogous
3
4
result was obtained by Gaspar and Galambosi when they
´ ´
analyzed the same problem under CSTR conditions in another
Belousov–Zhabotinsky system.
Fig. 4 One example of excitability of the preoscillatory (a) and (c), and
postoscillatory (b) and (d) stable quasi-steady states in the vicinity of the
bifurcation points. Cases (a) and (b) are experimental results; (c) and (d)
The perturbations of the preoscillatory and postoscillatory
stable quasi-steady states are performed in the vicinity of the
corresponding bifurcation points.
ꢁ2
ꢁ3
are numerical simulation. [MA]
0
= 1.60 ꢀ 10 mol dm (case (d) in
Fig. 1). The moments of perturbations are denoted by arrows. The
perturbations intensity amounted to 7.84 ꢀ 10ꢁ5 mol dmꢁ3 KBr.
3
.2 Numerical simulations
Although the mechanism of the BZ reaction has been inves-
tigated extensively, we had several problems when we wanted
to simulate the experimentally obtained results using the
6
,29,35–42
proposed models.
For example, the oscillations were
obtained, but the preoscillatory period, if existing in numerical
simulations, was very short. Therefore, we optimized one
version of these models, added one species and some reactions
and then tried to simulate evolution of the BZ reaction realized
in the close reactor. With this aim we used the reaction scheme
shown in Table 1. It is derived from the one proposed by
4
0
¨
Gyorgyi and Field and expanded by the reactions analogous
4
6
to the ones appearing in iodine oscillatory processes. Parti-
cularly, the reactions (R1), (R4)–(R10) are taken from the
4
0
¨
Gyorgyi, Field (GF) model referred to as model B of the BZ
system in a CSTR. In the mentioned model we added the
reversible reaction (R-1), the reaction (R12) due to evapora-
2
7,34,47,48
tion of bromine, discussed previously,
reactions [(R2), (R3), (R-3) and (R11)], as a consequence of
including the new species, Br O. Namely, Br O species,
mentioned previously in ref. 49, and the related reactions are
introduced in the model in analogy to I O that exists in the
and four new
Fig. 3 Period of time between two successive peaks (periods between
oscillations) as function of instantaneous malonic acid concen-
0
tration at the beginning of the corresponding period. [MA] =
2
2
ꢁ
2
ꢁ3
4
.30 ꢀ 10 mol dm (case (g) in Fig. 1). Solid and empty rhombuses
denote experimental and numerical values, respectively.
2
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Table 1 Reactions and their rate constants used in the numerical simulation
a
Reaction
Rate constants for 301 C
Ref.
43
ꢁ
+
Br + HOBr + H - Br
9
ꢁ2
6
ꢁ1
ꢁ1
ꢁ1
ꢁ1
R1
R-1
R2
R3
R-3
R4
R5
R6
R-6
R7
R-7
R8
R9
2
+ H
2
O
2.55 ꢀ 10 mol dm s
ꢁ
+
ꢁ1
Br
HBrO
Br O+H
2HOBr - Br
2
+ H
2
O - Br + HOBr + H
3.18 s
43
ꢁ
+
6
ꢁ2
6
c
2
+ Br + H - Br
O - 2HOBr
O+H
2
O + H
2
O
5.93 ꢀ 10 mol dm s
3
ꢁ1
c
c
2
2
3.21 ꢀ 10 s
8
ꢁ1
3
2
2
O
3.22 ꢀ 10 mol dm s
ꢁ
ꢁ
+
ꢁ3
3
9
2.86 mol dm s
ꢁ1
Br + BrO
3
+ 2H - HOBr + HBrO
2
23, 43
23, 43
43
ꢁ
+
ꢁ1
3
2HBrO
2
- BrO
+ HBrO
+ H
3
+ HOBr + H
3.49 ꢀ 10 mol dm s
ꢁ2 ꢁ1
44.70 mol dm s
ꢁ
+
ꢅ
6
BrO
2BrO
3
2
+ H - 2BrO
2
+ H
2
O
ꢅ
3ꢁ
+
7
ꢁ1
3
ꢁ1
ꢁ1
ꢁ1
2
2
O - BrO + HBrO
2
+ H
6.70 ꢀ 10 mol dm s
43
3+
ꢅ
+
4+
4
ꢁ2
6
b
Ce + BrO
2
+ H - Ce + HBrO
2
+
3.20 ꢀ 10 mol dm s
4+
3+
ꢅ
4
ꢁ1
3
Ce + HBrO
MA + Br
2
- Ce + BrO
2
++ H
1.12 ꢀ 10 mol dm s
23
ꢁ
ꢁ1
ꢁ1
3
4.24 mol dm s
ꢁ1
ꢁ1
b
2
4
- BrMA + Br + H
+
3+
+
3
0.36 mol dm s
MA + Ce - Ce + P
BrMA + Ce - Ce + Br + P
1
+ H
23
23
4+
3+
ꢁ
ꢁ1
3
ꢁ1
R10
R11
R12
2
47.17 mol dm s
4.23 ꢀ 10 mol dm s
ꢁ ꢁ1
s
ꢁ2
ꢁ1
3
ꢁ1
c
Br
Br
2
O + MA - BrMA + HOBr
(sol) - Br (g)
2
b
2
2
1.10 ꢀ 10
a
Abbreviations: MA = malonic acid, BrMA = bromomalonic acid. The concentration of water [H
ꢁ3
2
O] = 55 mol dm is included in the rate
constants. The rate constants at T = 25 1C are taken from ref. 23 and 43 and then recalculated by means of the activation energies from ref. 44 and
b
4
5 to the corresponding values at 30 1C. The starting values are taken from ref. 23 for reaction (R7), 39, 43, 45 for (R8) and 27 for (R12), and
c
modified in this work to fit the experimental data. The rate constants defined in this work.
4
6
model for Bray–Liebhafsky oscillatory reaction. Formally,
the second step of the basic model was replaced by three steps
references and then optimized in this work, with the aim of
reproducing the experimental behavior. The rate constants for
reactions (R2), (R3) and (R11) are determined here during
numerical simulations. Although all experiments were
[
reactions (R2), (R3) and (R-3)]. Such introducing of Br O
2
species as a new intermediate is appropriate for the fine
regulation of the HOBr concentration in the reaction system,
ꢁ
3
performed in 1.00 mol dm
+
H
2
SO , in all the simulations we
4
ꢁ3
since we assume that Br
bromomalonic acid and HOBr (R11). The involvement of
Br O in the explanation of the reaction mechanism can also
be found in the paper by Pelle et al. There, beside our
reaction (R3), the reaction between Br O and oxalic acid is
taken into account. The products of that reaction are Br , CO
2
O reacts with malonic acid to yield
0
use [H ] = 1.29 mol dm
according to Robertson and
5
1
Dunford. In the simulations, the concentration of water is
included in the corresponding rate constants. The concentra-
tions of other species are treated as dynamical variables.
The simulations based on the proposed model with rate
constants given in Table 1 are carried out under the conditions
used in the experiments (Fig. 1). The obtained time evolutions
have the same type and structures as the experimental ones.
The most important oscillatory features [preoscillatory period
2
4
9
2
2
2
and H O, whereas in our case the products of the reaction
2
between Br O and malonic acid are bromomalonic acid and
2
HOBr.
Preliminary numerical calculations based on the reactions
(t ), the time elapsed between the start of the reaction and the
1
(
R1)–(R11) can very well simulate the experimentally obtained
4
termination of the oscillatory phase (tend), the period of
oscillations and the number of oscillations (n)], are well
simulated. However, the calculated amplitudes of oscillations
are higher than the measured ones but they are similar to the
amplitudes obtained by GTF and other models derived from
0
oscillograms, better than the original model B, with the
correct number, form and structure of oscillations, but with
2–3 times shorter preoscillatory periods. The length of the
preoscillatory period depends predominantly on the concen-
5
0
41
tration of bromomalonic acid. Therefore, the reduction of
the bromine concentration caused by bromine evaporation
this one. Moreover, there is an obvious difference between
the period of the last incomplete oscillation in the experiment
and in the calculations.
2
7,34,48
from the system
The rate constant k calculated from eqn (2), where tend is
determined from the numerically simulated oscillograms, is
kevap
Br2ðsolÞ ꢁ! Br2ðgÞ
ðR12Þ
ꢁ
2
ꢁ1
equal to k (calculated) = 2.38 ꢀ 10 min . This value for k,
obtained numerically, is very close to the one obtained
experimentally.
would influence the reaction between bromine and malonic
acid (R8), and consequently lead to the prolongation of the
preoscillatory period. The oscillatory region begins when a
certain amount of bromomalonic acid BrMA is accumulated
in the reaction mixture.
The malonic acid concentrations at tend, as determined from
the numerically simulated oscillograms by a procedure analogous
to the treatment of the experimental results equal to [MA]end
ꢁ
=
3
ꢁ3
Almost all rate constants are taken from earlier numerical
simulations. However, some of them had to be modified for the
purpose of simulating oscillograms obtained under our experi-
mental conditions. Particularly, the values of the rate constants
for reactions (R1), (R4)–(R6), (R9)–(R10) are directly taken
6.80 ꢀ 10 mol dm , is higher than the one obtained experi-
ꢁ3 ꢁ3
mentally (6.02ꢀ 10 mol dm ). The critical initial malonic acid
concentration obtained when t = tend from numerically simu-
lated oscillograms, is equal to [MA]0,crit = 9.26 ꢀ 10 mol dm
The calculated critical malonic acid concentration [MA]crit, at the
1
ꢁ
3
ꢁ3
.
2
3,43
ꢁ3
ꢁ3
from literature,
(
whereas the rate constants of reactions
R7), (R8) and (R12) are initially taken from the mentioned
same point is 6.69 ꢀ 10 mol dm , whereas the value of
ꢁ3 ꢁ3
[MA]end is 6.80 ꢀ 10 mol dm (Table 2).
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662 | Phys. Chem. Chem. Phys., 2008, 10, 6658–6664

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Table 2 Experimental and numerical results for the rate constant of overall malonic acid decomposition and characteristic malonic acid
concentrations
ꢁ1
ꢁ3
ꢁ3
ꢁ3
crit/mol dm
k/min
[MA]end/mol dm
[MA]crit/mol dm
[MA]
0
,
ꢁ
ꢁ
2
2
ꢁ3
ꢁ3
ꢁ3
ꢁ3
Experimental results
Numerical results
2.14 ꢀ 10
2.38 ꢀ 10
6.02 ꢀ 10
6.12 ꢀ 10
9.20 ꢀ 10
9.26 ꢀ 10
ꢁ3
ꢁ3
6.80 ꢀ 10
6.69 ꢀ 10
The calculated periods between oscillations are given in
Fig. 3 (empty circles), whereas the simulation of perturbations
can be found in Fig. 4. Analyzing the oscillograms obtained by
simulation of the BZ reaction using the proposed model with
the parameters given in Table 1, the relaxation oscillations
with increasing periods between oscillations are found. More-
over, the excitability in the vicinity of the bifurcation points in
during the preparations of the article. The authors are grateful
to Mrs B. Meseldzija for her assistance in improving the
English text. This work was partially supported by the
Ministry for Science of the Republic of Serbia (Grants no.
142025 and 142019).
t
1
and tend is noticed. Hence, the SNIPER bifurcation points
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¨
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the model of the Belousov–Zhabotinsky reaction with Br O as
2
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