Chemical systems stationary states

In the present system, as in the one analyzed in the previous chapter there is a constant number of receptor molecules (no production and no degradation), switching between the available binding states. Therefore, the system stationary state should correspond to chemical equilibrium, and so ... 

What about the thermodynamic considerations As we have seen, processes that involve the transition of a fixed number of molecules between different states are compatible with thermodynamic equilibrium in the steady state, because all the states have the same chemical potential. The contrary happens with processes in which molecules are constantly produced and degraded. The synthesis and degradation of molecules implies the conversion of high energy substrates into low energy products, as well as the dissipation as heat of the energy difference. Furthermore, in order to maintain the system stationary state, new substrate molecules have to be... 

As already mentioned, a continual inflow of energy is necessary to maintain the stationary state of a living system. It is mostly chemical energy which is injected into the system, for example by activated amino acids in protein biosynthesis (see Sect. 5.3) or by nucleoside triphosphates in nucleic acid synthesis. Energy flow is always accompanied by entropy production (dS/dt), which is composed of two contributions ... 

In their subsequent works, the authors treated directly the nonlinear equations of evolution (e.g., the equations of chemical kinetics). Even though these equations cannot be solved explicitly, some powerful mathematical methods can be used to determine the nature of their solutions (rather than their analytical form). In these equations, one can generally identify a certain parameter k, which measures the strength of the external constraints that prevent the system from reaching thermodynamic equilibrium. The system then tends to a nonequilibrium stationary state. Near equilibrium, the latter state is unique and close to the former its characteristics, plotted against k, lie on a continuous curve (the thermodynamic branch). It may happen, however, that on increasing k, one reaches a critical bifurcation value k, beyond which the appearance of the... 

In our approach, we analyze not only the steady-state reaction rates, but also the relaxation dynamics of multiscale systems. We focused mostly on the case when all the elementary processes have significantly different timescales. In this case, we obtain "limit simplification" of the model all stationary states and relaxation processes could be analyzed "to the very end", by straightforward computations, mostly analytically. Chemical kinetics is an inexhaustible source of examples of multiscale systems for analysis. It is not surprising that many ideas and methods for such analysis were first invented for chemical systems. 

Fig. 1.14. Four more of the possible stationary-state bifurcation diagrams for chemical systems (see also Fig. 1.2) in flow reactors (a) isola (b) mushroom (c) isola + hysteresis loop ...

Chemical reactions with autocatalytic or thermal feedback can combine with the diffusive transport of molecules to create a striking set of spatial or temporal patterns. A reactor with permeable wall across which fresh reactants can diffuse in and products diffuse out is an open system and so can support multiple stationary states and sustained oscillations. The diffusion processes mean that the stationary-state concentrations will vary with position in the reactor, giving a profile , which may show distinct banding (Fig. 1.16). Similar patterns are also predicted in some circumstances in closed vessels if stirring ceases. Then the spatial dependence can develop spontaneously from an initially uniform state, but uniformity must always return eventually as the system approaches equilibrium. 

There is a period of time or, if we prefer, a range of reactant concentration over which the system spontaneously moves away from the pseudo-stationary state. The idea that stationary states may be unstable is not widely appreciated in chemical kinetics but it is fundamental to the analysis of oscillatory systems. 

These have only one true stationary-state solution, ass = pss = 0 (as t - go), corresponding to complete conversion of the initial reactant to the final product C. This stationary or chemical equilibrium state is a stable node as required by thermodynamics, but of course that tells us nothing about how the system evolves in time. If e is sufficiently small, we may hope that the concentrations of the intermediates will follow pseudo-stationary-state histories which we can identify with the results of the previous sections. In particular we may obtain a guide to the kinetics simply by replacing p by p0e et wherever it occurs in the stationary-state and Hopf formulae. Thus at any time t the dimensionless concentrations a and p would be related to the initial precursor concentration by... 

The traces in Fig. 3.9 were computed for a system with an uncatalysed reaction rate constant such that jcu > g, and hence there are no oscillatory responses in the corresponding pool chemical equations. For ku < we may also ask about the (time-dependent) local stability of the pseudo-stationary state. The concentration histories may become unstable to small perturbations for a limited time period. For sufficiently small e this should occur whilst the group p0e cr lies within the range... 

In this chapter we give an introduction and recipe for the full Hopf bifurcation analysis for chemical systems. Rather than work in completely general and abstract terms, we will illustrate the various stages by using the thermokinetic model of the previous chapter, with the exponential approximation for simplicity. We can draw many quantitative conclusions about the oscillatory solutions in that model. In particular we will be able to show (i)that the parameter values given by eqns (4.49) and (4.50) for tr(J) = 0 satisfy all the requirements of the. Hopf theorem (ii)that oscillatory behaviour is completely confined to the conditions for which the stationary state is... 

Another form of behaviour exhibited by a number of chemical reactions, including the Belousov-Zhabotinskii system, is that of excitability. This concerns a mixture which is prepared under conditions outside the oscillatory range. The system sits at the stationary state, which is stable. Infinitesimal perturbations decay back to the stationary state, perhaps in- a damped oscillatory manner. The effect of finite, but possibly still quite small, perturbations can, however, be markedly different. The system ultimately returns to the same state, but only after a large excursion, resembling a single oscillatory pulse. Excitable B-Z systems are well known for this propensity for supporting spiral waves (see chapter 1). 

Fig. 5.11. Excitability in a chemical system, (a) The nullclines /(a, 0) = 0 and g(a,0) = 0 intersect just to the left of the maximum. A suitable perturbation must make a full circuit, as shown by a typical trajectory, before returning to the stable stationary state, (b), (c) The corresponding evolution of the concentration of intermediate A and the temperature excess in time showing the large-amplitude excursion.

The CSTR is, in many ways, the easier to set up and operate, and to analyse theoretically. Figure 6.1 shows a typical CSTR, appropriate for solution-phase reactions. In the next three chapters we will look at the wide range of behaviour which chemical systems can show when operated in this type of reactor. In this chapter we concentrate on stationary-state aspects of isothermal autocatalytic reactions similar to those introduced in chapter 2. In chapter 7, we turn to non-isothermal systems similar to the model of chapter 4. There we also draw on a mathematical technique known as singularity theory to explain the many similarities (and some differences) between chemical autocatalysis and thermal feedback. Non-stationary aspects such as oscillations appear in chapter 8. 

For the no reaction state ass = a0, the relaxation time given by eqn (8.10) is simply equal to the residence time. In terms of the eigenvalue, we have A = - l/tres, which is negative. The stationary state is always stable, irrespective of a0 and kl. Chemistry makes no contribution (formally we have l/tch,ss = 0, so the chemical time goes to infinity) the perturbation of a does not introduce any B to the system, so no reaction is initiated. The recovery of the stationary state is achieved only by the inflow and outflow. 

In chapter 12 we discussed a model for a surface-catalysed reaction which displayed multiple stationary states. By adding an extra variable, in the form of a catalyst poison which simply takes place in a reversible but competitive adsorption process, oscillatory behaviour is induced. Hudson and Rossler have used similar principles to suggest a route to designer chaos which might be applicable to families of chemical systems. They took a two-variable scheme which displays a Hopf bifurcation and, thus, a periodic (limit cycle) response. To this is added a third variable whose role is to switch the system between oscillatory and non-oscillatory phases. 

A structure for a system is represented in quantum mechanics by a wave function, usually called a function of the coordinates that in classical theory would be used (with their conjugate momenta) in describing the system. The methods for finding the wave function for a system in a particular state are described in treatises on quantum mechanics. In our discussion of the nature of the chemical bond we shall restrict our interest in the main to the normal states of molecules. The stationary quantum states of a molecule or other system are states that are characterized by definite values of the total energy of the system. These states are designated by a quantum number, repre-... 

A much larger variety of phenomena can be described as stationary states of open chemical systems, i.e., systems in which molecules can be injected and from which molecules can be extracted. The simplest possibility for injection of molecules X is at a constant rate b. Extraction of X can only be done as long as there are some X present the rate of extraction must therefore vanish when n = 0 the simplest possible choice for it is an. Then the macroscopic equation for the number n of molecules X has the form... 

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... 

In 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

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