Thermodynamic approach in Chemical Complex Systems

Thermodynamic approach in Chemical Complex Systems Jacques Rieumont, R.

Quintana &

Phys.-Chem. Dept.Fac. of Chemistry,Havana University

[email protected]

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.



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 .


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.


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


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


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


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 .


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.


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.


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.


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.


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 )


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


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


k

k

CA 0 then Ck represent the concentration of the

product.


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


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


k

k

C

v

0 then Ck represent the concentration of the

product.


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 )


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 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.


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.

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Thermodynamic approach in Chemical Complex Systems