Untangling Complex Systems, page 63
The results are shown in Figure 9.39. The first situation originates pattern a with parallel
stripes. The second situation gives rise to pattern b with diagonal parallel stripes. The third
and fourth conditions originate the hexagonal patterns c and d, respectively. The activator
is abundant in the white regions of the graphs, whereas the inhibitor is abundant in the
black regions of the same graphs.
9.14. The differential equation describing how [ P] changes over time and space is:
∂[ P]
2
= −
[ P]
kd[ P] +
∂
D
∂ t
P
∂ r 2
(a)
(b)
(c)
(d)
FIGURE 9.39 Turing patterns obtained from the Schnackenberg’s model in the in the four conditions (a, b,
c, and d) listed in the text of the exercise.
310
Untangling Complex Systems
At the stationary state
k
2
d
[ P]
[ P] = ∂
D
2
P
r
∂
The solution will be a function of the type [ P] e br
= − , where b = kd DP . The analytical
solution suggests us that [ P] decays exponentially. The spatial distribution of the mor-
phogen depends on the ratio between the kinetic constant k , representing the reaction(s)
d
depleting P, and the diffusion coefficient D .
P
9.15. For the preparation of the stock solutions, we weight the salts, and we dissolve them
in deionized water. By performing such operations, we make systematic errors, and
we might also make random errors. The unavoidable systematic errors are due to the
use of the balance, the pipettes, and the flasks. For instance, for the preparation of the
KBrO solution in sulfuric acid and water, we weight (10.0200
;
3
± 0.0002) g of KBrO3
we dissolve them in (20.00 ± 0.03) mL of a previously prepared stock solution of H SO
2
4
(2.988 ± 0.036) M and, then, we add deionized water until we have (100.0 ± 0.1) mL. If
we avoid introducing any random error in our operations, the final concentrations of potas-
sium bromate and sulfuric acid in the stock solution are [H SO ]
2
4 = (0.598 ± 0.009) M
and [KBrO ]
3 = (0.6000 ± 0.0006) M, respectively. The uncertainties have been estimated
through the formula of maximum a priori absolute error (read Appendix D). The uncer-
tainty of [H SO ] is larger than that of [KBrO ] because the sulfuric acid solution was
2
4
3
prepared through two dilution steps.
The exercise requires to mix the reagents in three distinct conditions, maintaining
constant the total volume of the reactive solution. V
(12 0. 0. )
tot =
± 1 mL. Table 9.5 reports
the volumes of the solution A taken in the three distinct experiments. Moreover, it reports
the calculated final concentrations of KBrO
−
, H SO
, H+
,
3 ([BrO3 ] f )
2
4 ([H S
2 O4 ] f )
([H+] f )
and the product [H+] [BrO−
f
3 ] f . The values of [H+ ] f have been calculated by considering
a complete first dissociation of H S
2 O4, and, for the second dissociation, the equilibrium
constant K
2
−
a 2 = 1×10
M.
Figure 9.40 shows an example of the chemical waves that can be observed by per-
forming this exercise. Chemical waves do not behave like physical waves. In fact, when
one wave front runs into a wall of the Petri dish or when two wave fronts collide, we
do not see the phenomenon of reflection. A chemical wave can come back only if new
reagents are supplied in the path it traced. Chemical waves do not pass through each
other when they collide, but instead, they destroy each other because reagents have been
exhausted. The time required to replenish the system of new reagents is named as the
refractory period. Only after elapsing the refractory period, a new chemical wave can
pass through, again.
For each one of the three [H+] [BrO−
f
3 ] f values tested, we have measured several wave
velocities. Then, we have calculated their average and the standard deviation of the average
TABLE 9.5
Volumes of the Solution A and the Final Concentrations of BrO−3, H2SO4, H+ Used in the
Three Experiments
Volume of A (mL)
[BrO−
[H
[H+ ]
+
−
2
f (M)
[H ] f B
[ rO3 ] f M
(
)
2 SO4 ] f (M)
3 ] f M
( )
(7.0 ± 0.1)
(0.350 ± 0.008)
(0.349 ± 0.013)
(0.358 ± 0.019)
(0.125 ± 0.009)
(6.0 ± 0.1)
(0.300 ± 0.008)
(0.299 ± 0.012)
(0.308 ± 0.018)
(0.0924 ± 0.008)
(5.0 ± 0.1)
(0.250 ± 0.007)
(0.249 ± 0.011)
(0.258 ± 0.016)
(0.0645 ± 0.006)
The Emergence of Order in Space
311
FIGURE 9.40 A picture showing chemical waves.
(see Table 9.6). We have done the same for the determination of the period of the oscilla-
tions (see Table 9.6).
After plotting log( v) and log( T) versus log([H+] [BrO−
f
3 ] f ), we determine the exponents
n′ and n″ as the slopes of straight lines determined by the least squares method (see
Figure 9.41).
Note that the period of the oscillations shortens whereas the velocity of the chemical
waves grows up by increasing the product [H+] [BrO−
f
3 ] f . The linear relationship between
log( T) and log([H+] [BrO−
f
3 ] f ) is:
log( T ) = c′ + n′log +
−
H
BrO
+
−
(
3
f
f )
3
( .
0 4
.
0 )
2
( .137 .023)lo
(
)= ± − ±
g H
BrO
.
f
f
The correlation coefficient is r = 0 986
.
.
The linear relationship between log( v) and log([H+] [BrO−
f
3 ] f ) is:
log( v) = c′′+ n′′log([ +]
−
H
+
−
f [BrO3 ] f ) = (− .
0 8 ± .
0 )
2 +(1.22 ± .
0
)
18 log(([H ] f [BrO3 ] f), with
r = 0 989
.
.
The
estimated ′
n = −(1.37 ± 0. )
23 includes the value of the slope we found for the
relationship log( T ) = cost + n log([KBrO
([
])
3 ]) = ( .
0 32 ± .
0
)
05 + (− .
1 61± .
0 04)log KBrO3
in exercise 8.2. We may assume that ′
n corresponds to (−3/2) and ′′
n to (+1), within
TABLE 9.6
Velocities and Periods Measured as a Function of the [H+] B
[ rO−
f
3 ] f Product
v (cm/s)
log( v)
T (s)
log( T)
log H
([ + ] f Br
[ O− ] )
3 f
(0.0120 ± 0.0003)
(−1.92 ± 0.01)
(44.2 ± 0.6)
(1.646 ± 0.005)
(−0.903 ± 0.03)
(0.00755 ± 0.00006)
(−2.122 ± 0.003)
(76 ± 2)
(1.88 ± 0.01)
(−1.03 ± 0.04)
(0.00533 ± 0.00006)
(−2.273 ± 0.005)
(111 ± 1)
(2.045 ± 0.005)
(−1.19 ± 0.04)
312
Untangling Complex Systems
2.2
−1.8
2.1
−1.9
2.0
−2.0
T )
v)
1.9
log(
log(
−2.1
1.8
−2.2
1.7
−2.3
1.6
1.5
−2.4
−1.3
−1.2
−1.1
−1.0
−0.9
−0.8 −1.3
−1.2
−1.1
−1.0
−0.9
−0.8
log([H+]
−
−
f [BrO3 ] f)
log([H+] f [BrO3 ] f)
FIGURE 9.41 Linear trends of log( T) and log( v) versus log([H+ ] [BrO−
f
3 ] f ).
the uncertainties of our determinations. If we remind the definition of the velocity
of a wave as the ratio between its wavelength and its period ( v = (λ T )), we find that
λ = c′+ c′
+
−
n′+ n′′
10
([H ] f [BrO3 ] f )
. Therefore, ′
n + ′
n ≈ −(1/ )
2 , as expected.
1/2
According to Field and Noyes (1974b), v = ( kD +
−
4
[H ][BrO3 ]) . If we plot the data
of velocity vs. the products ([ +] [
−
1 2
f
3 ] f ) /
H
BrO
, we find that the slope of the straight line
(determined by the least squares method) is (
) /
4
1 2
(0.048 0.
)
009
1
1
kD
=
±
− −
cmM s . From the
latter result, it is possible to estimate the value of the kinetic constant: k = 28 ±
− −
(
)
11
2
1
M s .
The value k = 20 M−2 s−1 falls within the range determined in this exercise.
More accurate results can be achieved by eliminating convection in our experiments.
The Petri dish allows evaporation, causing temperature gradients between the top and
the bottom part of the solution and hence convection. There are two strategies to avoid
convection. A strategy is to minimize the thickness of the solution layer, maintaining
wet the entire surface of the Petri dish. The other strategy is to add a surfactant to the
solution. The risk of having convection can be eliminated if the BZ reaction is carried
out within a gel.
9.16. If we consider an expanding circle of a spiral wave, we have vc = 0 when vp = − c * D. It derives that r = ( D/ v )
p = .
0 0017cm ≈ 20 µm.
9.17. A fire spreading in a forest can be described as it were a chemical wave. Any forest is
an excitable medium. Therefore, the fire is an example of a trigger wave. It is peculiar
because the autocatalyst of the oxidation reactions responsible for the fire is the high local
temperature. The higher the temperature, the faster the oxidation reactions. Usually, the
refractory period for a fire is pretty long, because wherever it goes through, it leaves only
ashes. Therefore, new vegetation needs to be born before a new fire can pass, again.
The Emergence of Order in Space
313
9.18. Two properties are peculiar to chemical waves. One is the refractory period. The other is
the possibility of not having attenuation. If the fuel of the propagating chemical reaction is
distributed uniformly in the space, the chemical wave does not attenuate.
9.19. The precipitation of the red Ag CrO within the gels contained in cylinders and test tubes
2
4
starts within one hour or two after the addition of the silver nitrate solution. The precipita-
tion and the formation of the rings’ pattern continue to develop over a period that lasts even
more than twenty days. In Figure 9.42, there is a sequence of pictures taken at different
delay times. Picture A has been taken soon after the addition of the outer electrolyte solu-
tion over the gel. The other images refer to the same sample after 1 day (B), 4 days (C),
7 days (D), 13 days (E), and 27 days (F). It is evident that the number of rings increases
with time. Their formation starts at the interface between the solution of AgNO and the
3
gel. Then, day by day, always new rings develop towards the bottom part of the cylinder.
The direction of their appearance is opposite to that observed for the BZ reaction in exer-
cise 9.1. In fact, in the case of the BZ reaction, the rings form from the bottom part of the
solution, and they move towards the top.
Sometimes, the nucleation process is made evident by the formation of opaque yellow
layers of Ag CrO nanoparticles within the gel (see Figure 9.43).
2
4
For the verification of the time law xn ~ tn , we need to determine the distance L swept
by the front of the precipitation reaction at various delay time tn. By plotting L versus tn
(Figure 9.44), we obtain a straight line whose slope may be assumed to be D (with D
being the diffusion coefficient of silver nitrate).
The slope of the plot is s = ( .
± .
)
.
0 518 0 014
0 5
cm/h . It derives that
D = 0 27 ± 0 01
2 h = 7 5 ± 0 3 × −
( .
. )
( .
. ) 10 5
cm
cm2 /s. The diffusion coefficient of the silver
nitrate in the gel is of the same order of magnitude of its D in water.
The trends of the number of rings as a function of the time (expressed in hours) for the
two concentrations of AgNO are shown in Figure 9.45.
3
A way for estimating the average wavelength of the pattern ( λ) is to divide the distance
between the interphase layer and the farthest ring ( d) by the total number of rings ( N) (Table 9.7).
The wavelengths of the Liesegang patterns are of the same order of magnitude of the
wavelengths for the pattern generated by the BZ in exercise 9.1.
According to equation [9.10], the lower the concentration, the longer the wavelength.
The ratio between the two estimated wavelengths is
(A)
(B)
(C)
(D)
(E)
(F)
0 hours
1 day
4 days
7 days
13 days
27 days
FIGURE 9.42 Snapshots that show the formation of a Liesegang pattern within a test tube soon after the
addition of the outer electrolyte (A), and after 1 day (B), 4 days (C), 7 days (D), 13 days (E), 27 days (F).
314
Untangling Complex Systems
FIGURE 9.43 Temporary formation of thin layers of Ag CrO nanoparticles, indicated by the arrow.
2
4
12
9
m) 6
L (c
3
0
0
4
8
12
16
20
( t)½ ( hours)½
FIGURE 9.44 Trend of the distance L traveled by silver nitrate as a function of t 1/2 where t is the time elapsed.
λ
( .
0 60
.
0
)
01
R
dil
=
=
±
= (1.25 ± .
0
)
05
conc
λ
( .
0 48 ± .
0
)
01
R is close to the expected value of bconc / bdil = 2 ≈ 1 4
. .
If we use equation [9.10], it is possible to determine the kinetic constant value of the
deposition step [9.66] that is autocatalytic for the nuclei S:
4 2
π D
k =
= (0 011
.
±
−1 −
0 001
1
.
)M s
b 2
λ
The value of k for the deposition step [9.66] is an order of magnitude smaller than the value
of the autocatalytic step for the BZ reaction (see exercise 9.1). This result is the reason why
the formation of the final pattern in the case of the periodic precipitations takes more time
than in the case of the BZ reaction.
The Emergence of Order in Space
315
24
[AgNO3] = 0,589 M
21
[AgNO3] = 0,295 M
