Chemical reactions stationary states

Heterogeneous chemical reactions in which adsorbed species participate are not pure chemical reactions, as the surface concentrations of these substances depend on the electrode potential (see Section 4.3.3), and thus the reaction rates are also functions of the potential. Formulation of the relationship between the current density in the stationary state and the concentrations of the adsorbing species in solution is very simple for a linear adsorption isotherm. Assume that the adsorbed substance B undergoes an... 

If only the solvation of the gas-phase stationary points are studied, we are working within the frame of the Conventional Transition State Theory, whose problems when used along with the solvent equilibrium hypothesis have already been explained above. Thus, the set of Monte Carlo solvent configurations generated around the gas-phase transition state structure does not probably contain the real saddle point of the whole system, this way not being a correct representation of the conventional transition state of the chemical reaction in solution. However, in spite of that this elemental treatment... 

In a molecular mechanism there is always at least one step at which the system changes its initial identity, as it were, to acquire a different one. This is called here the chemical interconversion step. Following the viewpoint developed in our paper [43], the interconversion takes place unimolecular complex where the system jumps between quantum states having different stationary Hamiltonians. A simple reaction scheme is then one having only one such interconversion step. Chemical reactions proceeding with multiple interconversion steps can be treated along lines similar to the one step process as far as the quantum aspects are concerned. 

One common use of the stationary state approximation is with chain reactions. The simplest cases have three types of constituent chemical step, viz. chain initiation, chain propagation and termination. 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. 

The above models describe a simplified situation of stationary fixed chain ends. On the other hand, the characteristic rearrangement times of the chain carrying functional groups are smaller than the duration of the chemical reaction. Actually, in the rubbery state the network sites are characterized by a low but finite molecular mobility, i.e. R in Eq. (20) and, hence, the effective bimolecular rate constant is a function of the relaxation time of the network sites. On the other hand, the movement of the free chain end is limited and depends on the crosslinking density 82 84). An approach to the solution of this problem has been outlined elsewhere by use of computer-assisted modelling 851 Analytical estimation of the diffusion factor contribution to the reaction rate constant of the functional groups indicates that K 1/x, where t is the characteristic diffusion time of the terminal functional groups 86. ... 

In open, or flow, reactors chemical equilibrium need never be approached. The reaction is kept away from that state by the continuous inflow of fresh reactants and a matching outflow of product/reactant mixture. The reaction achieves a stationary state , where the rates at which all the participating species are being produced are exactly matched by their net inflow or outflow. This stationary-state composition will depend on the reaction rate constants, the inflow concentrations of all the species, and the average time a molecule spends in the reactor—the mean residence time or its inverse, the flow rate. Any oscillatory behaviour may now, under appropriate operating conditions, be sustained indefinitely, becoming a stable response even in the strictest mathematical sense. 

The concentration of A now tends to a value which makes the net inflow rate exactly balance the chemical reaction rate. When this has happened, da/dt = 0, so the concentration becomes steady. This stationary-state concentration can be maintained indefinitely. 

The stationary-state response curves, or bifurcation diagrams shown in Figs 1.13(b) and 1.12(f), represent two of the simplest possible patterns monotonic variation and a single hysteresis loop respectively. These are the only qualitatively different responses possible for the cubic autocatalytic step on its own. They are also found for a first-order exothermic reaction in an adiabatic flow reactor (see chapter 6). With only slightly more complex chemical mechanisms a whole array of extra exotic patterns can be found, such as those displayed in Fig. 1.14. The origins of these shapes will be determined in chapter 4. 

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. 

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... 

We have already determined the following information about the behaviour of the pool chemical model with the exponential approximation. There is a unique stationary-state solution for ass, the concentration of the intermediate A, and 0SS, the temperature rise, for any given combination of the experimental conditions /r and k. If the dimensionless reaction rate constant k is larger than the value e-2, then the stationary state is always stable. If heat transfer is more efficient, so that k Hopf bifurcation points along the stationary-state locus as /r varies (Fig. 4.4). If these bifurcation points are /r and /z (with the stationary state... 

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). 

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. 

If this value for a is substituted into the mass-balance eqn (6.3), it gives da/dt = 0. This is thus the stationary-state solution. Once it has been achieved the concentration remains constant. Note that at the stationary state the chemical reaction rate is not zero rather it is given by /c, ass, and this rate of conversion of A to B just balances the net rate of mechanical inflow of A to the reactor. 

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. 

Non-linearities arising from non-reactive interactions between adsorbed species will not be our main concern. In this section we return to variations of the Langmuir-Hinshelwood model, so the adsorption and desorption processes are not dependent on the surface coverage. We are now interested in establishing which properties of the chemical reaction step (12.2) may lead to multiplicity of stationary states. In particular we will investigate situations where the reaction step requires the involvement of additional vacant sites. Thus the reaction step can be represented in the general form... 

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