Untangling complex syste.., p.47
Untangling Complex Systems, page 47
FIGURE 8.16 Possible steady-state solutions for small (a), medium (b), and large (c) value of k .0
228
Untangling Complex Systems
the formula of maximum a priori absolute error (knowing that the KBrO molecular weight
3
is MW = (167.01 ± 0.01) g/mol):
∆
1
[
m
m
KBrO ]
m
( MW )
3
=
∆
∆
+ −
∆ V
0
2
2
( MW )( V )
+ − ( V)( MW)
( MW )( V )
6
= 6×10− + 2.1 10 5
−
2.63 10 4
−
.
2 9 10 4
×
+
×
=
× − mol/L
The largest contribution to the total uncertainty on KBrO
3
comes from
0
the determination of the volume. Finally, the value of the concentration is
KBrO
3
= (0.3503 ± 0.0003) mol/L. Similarly, we can estimate the uncertainties in the
0
concentration of the other stock solutions. After preparing the stock solutions, we mix spe-
cific amounts of them to start the BZ reaction. In the first run of the experiment, the initial
concentration of KBrO is:
3
[KBrO ]3 V ( .
M)( mL)
[KBrO
0 1
0 3503
2
]3 =
=
0.08544 mol/L
1
Vfin
(8 2. mL)
=
where:
V
stock solution that is taken and mixed with the other
1 is the volume of the KBrO3
reagents,
V is the total volume of the reaction mixture.
fin
The uncertainty in KBrO
3
is again estimated by the formula of maximum a priori abso-
1
lute error:
V
KBrO
3
KBrO V
∆ KBrO
1
0
3
V + −
0 1 ∆
3
∆ KBrO
=
3
∆
V
1
+
0
1
V
2
fin
fin
Vfin
Vfin
.
−
.
(0 3503)(2)
5
0 3503
=
mol
7 3
. 2 ×10 +
0.016 + −
0 070
.
=
0 0015
.
2
8.2
(
8.2)
L
Therefore, the concentrations of KBrO in the three runs of the reaction are:
3
[KBrO ] ( .
.
)
3
= 0 0854 ± 0 0015 mol/L
1
[KBrO ] (0 064
.
0.
)
3
=
± 001 mol/L
2
[KBrO ] ( .
.
)
3
= 0 0427 ± 0 0007 mol/L
3
Then, we calculate the logarithm of this quantities. The uncertainty in log KBrO
3
is
i
determined by the maximum a priori absolute error’s formula:
log e
∆(logKBrO
10
i ) =
3
∆ KBrO
KBrO
3
i
3
i
Figure 8.17 reports some spectra of the BZ reaction recorded in the absence (a) and the pres-
ence (b) of the indicator ferroin. In (a), it is evident that the solution is uncolored when the
chemical system is in its reduced state (i.e., when Ce+3 is the dominant form of the cerium
ions), and it becomes slightly yellow when Ce+4 becomes the dominant form. In the presence
of the redox indicator (see plot b), the solution is red when the coordination compound is in its
reduced state (with iron in the Fe+2 state) because it absorbs all the visible wavelengths except
those of the red region. When the solution is in its oxidized state, the complex ferriin (having
the iron ion in its Fe+3 state) absorbs all the visible wavelengths except those of the blue region.
The Emergence of Temporal Order in a Chemical Laboratory
229
1
Ce+4
A
Ce+3
0
300
400
500
600
700
800
(a)
λ (nm)
2
Fe+2
A 1
Fe+3
0
300
400
500
600
700
800
(b)
λ (nm)
FIGURE 8.17 Spectral evolution of the BZ reaction with the cerium ions as catalysts and in the absence
(a) and the presence (b) of ferroin as the redox indicator.
The profiles of the absorbance at 511 nm as a function of the time are plotted in
Figure 8.18 for the three different concentrations of KBrO , labeled as 1 (in a), 2 (in b), and
3
3 (in c), respectively.
The traces have been recorded for at least 30 minutes. From the kinetics, we determine
the periods of the oscillations. Then, we calculate the log T. We will have as many log T
as are the oscillations in each run. The best estimate will be the average, and its uncer-
tainty will be the standard deviation of the mean. Finally, we plot log( T ) ∆
( )
i ±
log Ti versus
log([KBrO ) ∆
(
) and we determine the exponent n by fitting the data with
3 ] i ±
log [KBrO3] i ,
the least-squares method, as shown in Figure 8.19.
It results that log( T ) =cost + n log([KBrO
(
)
3]) = (0.32 ± .
0 05)+(− .
1 61± 0.04)log [KBrO3]
with a correlation coefficient r = 0 99972
.
. The lower the [KBrO ], the longer the period.
3
The exponent is roughly –(3/2). This result is consistent with a mechanism in which bro-
mate plays a role in the rate-limiting steps.
8.3. During the progress of the BZ reaction, changes in the concentrations of protons, bro-
mate, malonic acid, and hypobromous acid are small because the amount of catalyst
used is almost two orders of magnitude smaller. Therefore, assuming that [H+], [BrO−3],
[CH
and [HBrO] remain unchanged during the reaction, it is possible to write
2 (COOH)2 ],
the following system of three nonlinear differential equations:
d X
= k [ H+ 2] A Y k
2
2
H+ X Y
k 3 H+
1
A Z
2 k 4 X
dt
−
+ −
d Y
= −
+
k
2
1[ H ]
A Y − k +
2 H X Y
fk 5 O Z
dt
+
d Z
= k
3 H +
A Z
k 5 O Z
dt
−
230
Untangling Complex Systems
1.5
A
600
900
1200
1500
1800
(a)
t (s)
1.5
A
600
900
1200
1500
1800
(b)
t (s)
1.5
A
600
900
1200
1500
1800
(c)
t (s)
FIGURE 8.18 Trends of A recorded at 511 nm for the three distinct concentrations of KBrO : (1) 0.0854 M
3
in a; (2) 0.064 M in b; (3) 0.0427 M in c.
2.6
2.5
2.4
T) 2.3
log( 2.2
2.1
2.0
–1.40 –1.35 –1.30 –1.25 –1.20 –1.15 –1.10 –1.05
log ([KBrO3])
FIGURE 8.19 Linearization of the relation between log( T ) and log([KBrO )
3 ] .
The function file to integrate the system of differential equations in MATLAB should look
like this:
function dy = Oregonator(t, y)
dy = zeros(3,1);
options=odeset(‘RelTol’,10^-6,’AbsTol’,10^-10);
k1 = 1.28;
k2 = 800000;
k3 = 8;
The Emergence of Temporal Order in a Chemical Laboratory
231
k4 = 2000;
k5 = 1;
A = 0.12;
O = 0.1;
f = 0.6;
dy(1)=k1*A*y(2)-k2*y(1)*y(2)+k3*A*y(1)-2*k4*y(1)*y(1);
dy(2)=-k1*A*y(2)-k2*y(1)*y(2)+f*k5*O*y(3);
dy(3)=k3*A*y(1)-k5*O*y(3);
where
k 1 is k
+ 2
+
+
1[ H ] , k 2 is k 2[ H ], k 3 is k 3[ H ], y(1) is X, y(2) is Y, y(3) is Z. The script file should look like the following:
[ ] =
(‘
, [0 1200], [0.0001 0.0001 0.002])
t, y
ode15s Oregonator’
wherein [0 1200] is the time window, whereas [0.0001 0.0001 0.002] are the initial condi-
tions for y(1), y(2), and y(3), respectively. The profiles of how the concentrations of X, Y
and Z change over time (calculated in the case of A = 0.18) are shown in Figure 8.20. It is evident that when the concentration of the activator X is high, then the concentration of
inhibitor Y is very low, and vice versa. This behavior is due to the coproduct autocontrol,
which is the time needed to store Z that finally regenerates Y.
The three-dimensional plot of Figure 8.21 shows that the chemical system approaches
a limit cycle whatever is the initial condition. Moreover, if we look at the projection on the
XY plane, we confirm that X and Y exclude reciprocally.
To determine the average value of the period of the oscillations we use the Fourier
transform of the trend of [ Z] versus time. An example of Fourier Transform is shown in
Figure 8.22.
The uncertainty on the frequency is given by the Half Width at Half Maximum of
the peak with the largest amplitude (see Figure 8.22). The period is given by T = 1 ν .
The uncertainty of the period is calculated by using the formula of the propagation of the
“a priori maximum absolute error”:
∆ T = − 1 ∆ν
2
ν
[ Z]
[ X]
[ Y]
0.002
[ X] and [
0.0002 Y
[ Z] (M)
] (M)
0.001
0.000
0.0000
550
600
650
700
Time (s)
FIGURE 8.20 Trends of [ Z], [ X] and [ Y] over time, when [ A] = 0.18 M.
232
Untangling Complex Systems
0.0025
0.0020
0.0015
[ Z]
0.0010
0.0000
0.0005
0.0001
0.0000
[ Y ]
0.0000
0.0001
0.0002
0.0002
0.0003
[ X]
FIGURE 8.21 Three-dimensional phase space of the Oregonator model.
0.0010
0.0008
0.0006
Width at half maximum
0.0004
Amplitude
0.0002
0.00000.00
0.02
0.04
0.06
0.08
0.10
v (Hz)
FIGURE 8.22 Fourier transform of the time series [ Z] versus time when [ A] = 0.18 M.
The numerical integration of the system of three differential equations and the determi-
nation of the period of the oscillations after the calculation of the Fourier Transform are
repeated for all the conditions proposed in the text of this exercise. Finally, to establish the
exponential term n of the relation T
n
∝
−
[BrO3 ] , it is convenient to build a log-log plot, like
that shown in Figure 8.23. The straight line determined by the least-squares method has a
slope n = −(0 6
. ± 0. )
1 .
8.4. The function file to integrate the system of three differential equations in MATLAB should
look like that presented in exercise 8.3. The values of the kinetic constants at the three tem-
peratures proposed in the text of the exercise are reported in Table 8.8.
They have been calculated through the Arrhenius equation ( k T
( Eact R( T 0 T
− i
( ) (
))
0 ))
i
k T
= e
.
The script file to solve this exercise should look like the following one:
[t y]=ode15s(‘Oregonator’,[0 1200], [0.0001 0.0001 0.002]);
The Emergence of Temporal Order in a Chemical Laboratory
233
2.05
2.00
1.95
T) 1.90
log( 1.85
1.80
1.75
–1.1
–1.0
–0.9
–0.8
–0.7
log([KBrO3])
FIGURE 8.23 Linearization of the relation between log( T) and log([KBrO ]).
3
TABLE 8.8
Values of the Kinetic Constants for the Elementary Steps Appearing in the Oregonator
Model
T (K)
k1 (mol−3dm9s−1)
k2 (mol−2dm6s−1)
k3 (mol−2dm6s−1)
k4 (mol−1dm3s−1)
k5 (mol−1dm3s−1)
293.15
2
1,000,000
10
2000
1
303.15
4.16
1,402,878
22.5
4758
2.58
313.15
8.24
1,925,972
48.2
10709
6.27
wherein [0 1200] is the time window, whereas [0.0001 0.0001 0.002] are the initial condi-
tions for y(1), y(2), and y(3), respectively. Feel free to change slightly the initial conditions.
The calculated average values of the periods are 79 s (at 293.15 K), 30 s (at 303.15 K),
and 13 s (at 313.15 K). If we plot ln(1/∆ T) versus 1/ T and we determine the fitting straight
line with the least-squares method, we obtain a slope of (−8300 ± 130), which corresponds
to E
(69000 1000)J mol (69 )
act =
±
=
±1 kJ / mol.
8.5. The concentrations of the reactants in the proposed five experiments are listed in Table 8.9
(such values have been inspired by the demonstration on chemiluminescence by Prypsztejn
et al. [2005]).
TABLE 8.9
Concentrations of the Reagents for the Orbán Reaction in Five Distinct Experiments
C(H2O2) M
V(KSCN) M
V(Luminol) M
V(NaOH) M
V(CuSO4) M
Exp. 1
0.2
0.03
7.4·10−4
0.02
2.4·10−4
Exp. 2
0.2
0.03
1.48·10−3
0.04
1.2·10−4
Exp. 3
0.2
0.06
7.4·10−4
0.02
1.2·10−4
Exp. 4
0.2
0.03
7.4·10−4
0.04
1.2·10−4
Exp. 5
0.64
0.03
7.4·10−4
0.02
2.4·10−4
234
Untangling Complex Systems
Exp. 1
3.0 × 108
0.0
0
1000
2000
3000
9.0 × 108
Exp. 2
6.0 × 108
3.0 × 108
0.0
0
1000
2000
3000
1.2 × 109
Exp. 3
9.0 × 108
6.0 × 108
intensity 3.0 × 108
ed
0.0 0
500
