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Jacques Rieumont, R. Quintana & Phys.-Chem. Dept. Fac. of Chemistry, Havana University [email protected]. José M. Nieto Villar. Thermodynamic approach in Chemical Complex Systems. Outline. The thermodynamic background. - PowerPoint PPT Presentation
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Thermodynamic approach in Chemical Complex Systems Jacques Rieumont, R.
Quintana &
Phys.-Chem. Dept.Fac. of Chemistry,Havana University
José M. Nieto Villar
Outline
The thermodynamic background. The rate of entropy production as
a discriminate function of the most important steps of chemical complex mechanism.
The rate of entropy production as a Lyapunov function in oscillating and chaotic chemical reactions.
1. Thermodynamic background
Af ter De Donder the entropy of the system dSs is defined
by
ies SSdS (1)
Where eS is the flow of entropy due to interactions with the exterior, and iδS is the entropy production due to
irreversible processes, namely chemical reactions, mass transport, i.e. According to the second law of thermodynamics, the entropy production is always positive or zero in equilibrium state, 0iS . The Gibbs f ree energy G is the thermodynamic potential used to describe the evolution of the chemical reaction, if the temperature and pressure are the constant, because it measure the entropy production of the systems by
TTPdG
iS (2)
Further on, we shall of ten make use of entropy production per unit time
dtdG
TdtS TPi 1 (3)
Where
pi SdtS is the rate of entropy production. But the Gibbs f ree energy is
also f unction of the extent of reaction . Considering T and P constant, the rate of entropy production
pS
dtdG
TS TPp
1 (4)
De Donder has called
TPG the affi nity of the chemical reaction. I t may
be expressed by
k
k
c
C
KRT log (5)
Where Kc is the Guldberg-Waage constant, R the gas constant, k the stoichiometric coeffi cients, and Ck the molar concentration.
1. Thermodynamic background
1. Thermodynamic background
The second term of (4)
dtd
is the rate of reaction V. According to
mass action law takes the f orm
kk
kiki CkCkVVV (6)
Taking into account (5) and (6) substituting in (4), we get
i
in
iiip
VV
VVRS log1
(7)
For oscillating or chaotic chemical reactions we can take the average
of
pS .
2. The rate of entropy production as a discriminate function of the most important steps of chemical complex mechanism
The most studied example of the occurrence of complex behaviors in chemical systems out of thermodynamic equilibrium has been provided by Belousov-Zhabotinsky reaction. The most complete mechanism was report by Gyorgy et al., in , and is know as GTF, it includes 80 reaction steps and 26 species, as shown in fig. I .
2. The rate of entropy production as a discriminate function of the most important steps of chemical complex mechanism
A sensitivity technique was applied to the to GTF mechanism given a subset of 42 reaction steps and 22 components that still reproduce the oscillating behavior of the original model.
We are concerned here with the question of the entropy production rate of each reaction steps as a sensitivity tool to discriminate the most important steps of a mechanism.
2. The rate of entropy production as a discriminate function of the most important steps of chemical complex mechanism
Brief comments about sensitivity analysis Let us consider a reaction mechanism that involves n species. By the law of Mass Action , we may f ormulate the associated kinetic equations, as a system of ordinary diff erential equations, ODE,
Where is the parameter vector and C is is the vector of intermediary concentrations.
(8) C,fdt
dC
2. The rate of entropy production as a discriminate function of the most important steps of chemical complex mechanism
Let us write the solution of (8) as f unction of time, initial concentrations and parameters, C = C(C0,,t) Then, sensitivity coeffi cients may be calculated by
Where (C,,t) is an nxm matrix depending upon initial conditions, time and parameters. The element of this matrix measures the local sensitivity of the system.
(9) ,,,0 tC,tCC
2. The rate of entropy production as a discriminate function of the most important steps of chemical complex mechanism
As we have seen the rate of the entropy production can be obtained by (7)
i
in
iiip
VV
VVRS log1
Let us consider a chemical reaction whose mechanism consists of k intermediary species and n steps. Then the rate of the entropy production f or the m-th step is
(10) log
m
mmmp
VV
VVRS m
We shall postulated that step m is dominant f or a given condition if
nSS nm pp
2. The rate of entropy production as a discriminate function of the most important steps of chemical complex mechanism
Although systems are expected to evolve f ar f rom thermodynamic equilibrium, we propose to use a criterion based on entropy production rate of reaction mechanism steps, that as a matter of f act, proof s to be useful to match experimental data. I ndeed, using entropy production rate, a so-called by us Method of the Dominating Steps, the original GTF model could be reduced down to 26 reaction steps and 20 species (see Table 1). This drastic reduction, on the one hand, incorporates the f ull richness of the pioneering work of Field et al. (FKN model), and, on the other hand, it is enough to account, in particular, f or the experimental results reported by Ruoff .
2. The rate of entropy production as a discriminate function of the most important steps of chemical complex mechanism
Figure 2 shows time series generated f or various values ofthe control parameter (bromate concentration). Theseresults agree quite well with the experimental fi ndingsreported by Ruoff . I n agreement with Ruoff we see thatthe chaotic behavior occur when the system approaches anexcitable steady-state.
2. The rate of entropy production as a discriminate function of the most important steps of chemical complex mechanism
3. The rate of entropy production as a Lyapunov function in oscillating and chaotic chemical reactions
The stability of a dynamic system, as it is known in the literature can be treated local and globally. From the analytic point of view, f or the analysis of the local stability, is used the qualitative theory, elaborated by Poincaré. The global stability, on the other hand, uses the theory elaborated by Lyapunov, through the so-called f unction of Lyapunov. On the one hand, the appropriate choice of this f unction depends in particular on the dynamic system under study. On other hand, there is no general method which set how to select the Lyapunov function f or a given dynamic system.
3. The rate of entropy production as a Lyapunov function in oscillating and chaotic chemical reactions
From the thermodynamic point of view, Prigogine demonstrated that in the linear region, the rate of entropy production is a Lyapunov function. A rigorous treatment, in connection with the global stability of the stationary states in the linear region, was developed parallelly by Katchalsky.
Far from the thermodynamic equilibrium, the global stability in the non-linear region, for example in the chemical reactions, is a topic that has not been resolved yet.
3. The rate of entropy production as a Lyapunov function in oscillating and chaotic chemical reactions
The Lyapunov Function in dynamical systems according with the theorem of Lyapunov about the stability, if a derivable f unction V(x1, x2,..., xn) called f unction of Lyapunov that satisfies, in an environment the f ollowing conditions: 1. V(x1, x2,..., xn) 0, and V= 0 if xi = 0 (i=1, 2,..., n) that is to say, the f unction V has a strict minimum in the origin of coordinated.
2.
n
ini
i
xxxfxV
dtdV
121 0,,,, as t t0, then the
stationary state xi is stable.
3. The rate of entropy production as a Lyapunov function in oscillating and chaotic chemical reactions
To generalize our ideas, let us consider a hypothetical complex chemical system, f ormed by k reaction steps, such as:
R1 = X1 R1 + X1 = X2 . . . . . . . . . Xk = Pk
Where R1, R2, …, Rk represent the concentration of reagents, P1, P2,…, Pk represent the concentration of the products, finally, X1, X2,…, Xk are the concentration of the intermediates of the reaction respectively.
3. The rate of a entropy production as a Lyapunov function in oscillating and chaotic chemical reactions
The rate of the entropy production of the system under consideration can be set as:
kk
k
ii
i AvTdt
SS
dt
S
1 (I )
This is always posit ive by virtue of the second Law of the thermodynamic one. Taking the Eulerian derivative of (I ), we get:
dtdA
vAdtdv
dtdA
vAdtdv
dtSd
dtSd k
kkk
k
kii
111
1
1
(I I )
3. The rate of entropy production as a Lyapunov function in oscillating and chaotic chemical reactions
The affi nity Ak is a f unction of the concentration of the reagents Ck or the products Pk of the reaction, the term
dtdAk can
be developed by means of the chain rule, and we get:
dtdC
CA
dtdC
CA
dtdC
CA
dtdA k
k
kk
2
2
21
1
1
3. The rate of entropy production as a Lyapunov function in oscillating and chaotic chemical reactions
We have this way that:
dt
dCk 0 if Ck is the concentration of reagent, on the
contrary we have that if :
dtdCk 0 then Ck represent the concentration of the
product. On the order hand, the term,
k
k
CA changes in the inverse
ration of the previous one, thus:
k
k
CA 0 if Ck is the concentration of reagent, on the
contrary we have that if :
k
k
CA 0 then Ck represent the concentration of the
product.
3. The rate of entropy production as a Lyapunov function in oscillating and chaotic chemical reactions
As can be seen, the term dtdAk , it is always negative
and does not depended of the f act the control parameter be a product or reagent.
Similarly, the term dtdvk can be developed by means
of the chain rule as a f unction of the concentration of the reagents Ck or the products Pk of the reaction, as
dtdC
Cv
dtdC
Cv
dtdC
Cv
dtdv k
k
kk
2
2
21
1
1
3. The rate of entropy production as a Lyapunov function in oscillating and chaotic chemical reactionsWe get:
dt
dCk 0 if Ck is the concentration of reagent, on the
contrary we have that if :
dt
dCk 0 then Ck represent the concentration of the
product. On the other hand the term,
k
k
C
v
it changes to the
inverse of the previous one, we have this way that:
k
k
C
v
0 if Ck is the concentration of reagent, on the
contrary we have that if :
k
k
C
v
0 then Ck represent the concentration of the
product.
3. The rate of entropy production as a Lyapunov function in oscillating and chaotic chemical reactions
Thus, the remaining terms of the equation (I I ), kk Adt
dv
are always negative, then the f ollowing condition is f ullfi lled:
01
k
kii
dtSd
dtSd (I I I )
3. The rate of entropy production as a Lyapunov function in oscillating and chaotic chemical reactions
I n order to illustrate our ideas, we shall consider the f ollowing model proposed by Rössler: A + x = 2x x + y = 2y B + z = 2z C + y = D X + z = E As a control parameter was take the concentration of reactant A. I n Figure 1 is shown the dependence of the rate of entropy production with the value of control parameter A.
3. The rate of entropy production as a Lyapunov function in oscillating and chaotic chemical reactions
3. The rate of entropy production as a Lyapunov function in oscillating and chaotic chemical reactionsTake the Eulerian derivative of the rate of entropy production as f unction of A, we get:
0
dtdA
A
S
dt
Sd pp
As is shown in figure A
S p
0. The term, dtdA
as a
consequence of the f act that A is a reagent, then
0dtdA . I n this way it is shown that, at least f or
chemical reactions, the rate of entropy production has proven to be a Lyapunov Function.
3. The rate of entropy production as a Lyapunov function in oscillating and chaotic chemical reactions
1. Nieto-Villar, J .M., García, J .M., Rieumont, J ., Entropy productionrate as an evolutive criteria in chemical systems. I . Oscillatingreactions, Physica Scripta, 1995, 52, 30.
2. García, J .M., Nieto-Villar, J .M., Rieumont, J ., Entropy productionrate as an evolutive criteria in chemical systems. I I . Chaoticreactions, Physica Scripta, 1996, 53, 643.
3. Rieumont, J ., García, J .M., Nieto-Villar, J .M., The rate of entropyproduction as a mean to determine the mos important reactionsteps in BZ reaction, Anales de Química, 1997, 4, 93.
4. Nieto-Villar, J .M., The Lyapunov Function in Chemical Systems,Anales de Química, 1997, 4, 93.
5. Nieto-Villar, J .M. & Velarde M.G., Chaos and Hyperchaos in aModelo f the Belousov-Zhabotinsky Reaction in a Batch Reactor, J .Non-Equilib. Thermodyn., 2000, v. 25, 269.