TIME ORDER - CHEMICAL OSCILLATIONS

Frontiers of non-equilibrium physical chemistry are continuously advancing on account of increasing interest in the study of far equilibrium region in real systems such as living systems, socio-political system, finance and market economy. In the earlier part, we had discussed the development and nature of steady states, close to equilibrium and their mode of transition to multiple steady states. 

Equilibrium stmctures are maintained without any exchange of energy and/or matter. A crystal is the prototype of an equilibrium structure. Dissipative structures, on the other hand, are maintained through exchange of energy (and in some cases also exchange of matter) with the outside world. Since entropy is dissipated in open systems under specific circumstances, emerging structures in such systems are called dissipative structures. 

In this chapter, it is intended to discuss more complex phenomena as we further move away from equilibrium. Even through d5 0 for such open systems, on account of export of entropy to the surrounding, order can be established with respect to time and space as observed. Further, in more complicated reaction-diffusion system, exotic dissipative structure such as Fractals can appear. 

Oscillatory reactions are a typical class of phenomena, which display unusual features. After the discovery of Belousov-Zhabotinskii (B-Z) reaction, there has been a tremendous flurry of activity [1] and a large number of such reactions have been discovered during recent years. Biochemical reactions [2-10] such as glycolytic oscillations and peroxidase catalysed oxidation of nicotinamide adenosine deoxyhydrogenase (NADH) have also generated considerable interest. The interest in such reactions is stiU sustained in view of their importance in understanding cardiac and neuronal oscillations. In the case of many oscillatory chemical reactions [1], detailed reaction mechanisms have been postulated and verified with the help of numerical computation. This has also been particularly so for B-Z reaction where Field-Koros-Noyes (FKN) mechanism [11] has been invoked. 

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This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

The autocatalytic reaction scheme A + 2B —> 3B, B —> C was introduced in 1983s and has proved itself to be fecund of useful applications in the study of reactor stability and chemical oscillations.6 We shall depart from their notation for we wish to be able to generalize to several species, Au and it is not desirable to use the concentration of A as a reference concentration when it is going to be varied. Similarly, the several species will have different rate constants for the several autocatalytic steps and therefore the first-order rate constant of B — C is most apt for the time scale. 

Example 13.6 Order in time Thermodynamic conditions for chemical oscillations Consider the following set of reactions... 

Complexity of the catalytic process itself. The catalytic processes are very complicated. One of the factors that influences catalyst properties includesnon-linearity of surface catalytic reactions which is rarely taken into considerations. The catalyst surface has a feature of fractional-dimension structures where the distributions of the active center on surface show multi-fractional-dimension characteristics. At the same time, there is a non-equilibrium phase change and space-time ordered structures such as the chemical oscillation and chaos during a certain process. 

In particular, it is useful to define the critical point through F(nc) = 0 (the stationary state). Since multicomponent chemical systems often reveal quite complicated types of motion, we restrict ourselves in this preliminary treatment to the stable stationary states, which are approached by the system without oscillations in time. To illustrate this point, we mention the simplest reversible and irreversible bimolecular reactions like A+A —> B, A+B -y B, A + B —> C. The difference of densities rj t) = n(t) — nc can be used as the redefined order parameter 77 (Fig. 1.6). For the bimolecular processes the... 

What is described above is an idea of the so-called chemical clock, that is a reaction with periodic (oscillating) change of reactant concentrations its period could be estimated as 5t > nc/p. In the condensed matter theory a leap in densities is interpreted as phase transitions of the first order. From this point of view, the oscillations correspond to a sequence in time of phase transitions where the two phases (i.e., big clusters of A s containing inside rare and small clusters of B s and vice versa) differ greatly in their structures. 

It is interesting to consider the shapes of the subharmonic trajectories that lock on the torus in the various entrainment regions of order p/q. The subharmonic period 4 at the 4/3 resonance horn is, for example, a three-peaked oscillation in time [Fig. 7(a)] and has three closed loops in its phase-plane projection [Fig. 7(b)], while the subharmonic period 4 at the 4/ 1 resonance is a single-peaked, single-loop oscillation [Figs. 7(d) and 7(e)]. A subharmonic period 2 at the 2/3 resonance is also included in Figs. 7(g) and 7(h). Multipeaked oscillations observed in chemical systems (Scheintuch and Schmitz, 1977 Flytzani-Stephanopoulos et al., 1980) may thus result from the interaction of frequencies of local oscillators. Such trajectories are the nonlin-... 

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