Kinetic trajectories

The section addresses the problem of specifying constraints (10), (16), (27), (32), and (40) on macroscopic kinetics as applied to various problems. Formalization of these constraints as well as constructions of MEIS are on the whole based on the Boltzmann assumption on the equilibrium of "kinetic" trajectories of motion toward point xeq and the possibility to describe them by autonomous equations of the form x - f(x). 

When setting the constraints on macroscopic kinetics in MEIS the idea of tree is useful even from the viewpoint of interpreting the applied method for formalization of these constraints. It (the idea) can help represent even the deformation of the region of feasible solutions in the thermodynamic space and the deformation of extremely simple representation of this region (a thermodynamic tree), and the projection of limited kinetic trajectories on the tree. In other words the use of the tree notion helps reveal the interrelations between kinetics and kinetic constraints, on the one hand, and thermodynamics, on the other. 

The main point of the method consists in determining the kinetic trajectory for responses of some reaction output parameter on the variation of rate constants of individual steps. The output parameter frequently implies the concentration of species of the reaction system or the reaction parameters for which the fimction type for the species concentration is known. Conceptually, via this procedure kinetic presence of an individual step in the reaction system is identified. 

Thus, the kinetic trajectories of sensitivities for concentration of reaction species (c ) may be determined by the rate constants of all the steps. For this, as we have made sure, the equation system needs to be solved with the number of equations equal to the munber of individual steps where each of these equations, in turn, includes 2m differential equations. 

The basic reason of difficulties that arise at conceiving the sensitivity kinetic trajectories is due to the fact that the sensitivity parameters specify the kinetic significance not strictly quantitatively, but for the most part qualitatively. The matter is that sensitivities of output parameters of a complex reaction, with respect to change in the values of rate constants of individual steps, characterize the influence of the rate for this step only indirectly and, consequently, its kinetic participation in the overall process. The rate of they-th step (rj) is equal to the product of the reaction rate constant and the concentrations of species participating in this step. So, for the bimolecular reaetion rj = kjCj c,. It is reasonable that even if one of the individual... 

The basis of the value approach is a Hamiltonian systematization of the mathematical model of a conq)lex reaction with the extraction of the characteristics (functional) of a target reaction and with the kinetic comprehension of conjugate variables. Regardless of a reaction s conplexity, pivotal in this approach is a universal numerical determination of kinetic trajectories of value units (the kinetic significance) of individual steps and chemical species of a complex reaction, according to the selected target reaction characteristic. 

Figure 3.6. Kinetic trajectories of value contributions of individual steps for the target functional I. Curves are numbered according to steps in Table 3.2.

Very useful information is provided by the calculated kinetic trajectories of the value contributions of individual steps, for the cases when a stimulator or inhibitor with optimum molecular structure participates in a reaction. Ranking the steps by the contribution of reactions with participation of stimulators, inhibitors and their intermediates, enables to answer the question what reaction or reactions namely play the dominant role, and finally to determine the structure of the most efficient stimulator or inhibitor. 

Numerieal methods in ehemieal kineties are used traditionally to calculate the kinetic trajectories of eoneentrations of the speeies of complex (multistep) chemical reactions, on the basis of their mathematical models. In particular, such models are then corrected after comparisons are made between the experimental data and compntational resnlts. 

Then calculations by equations (6.4) result in kinetic curves for benzaldehyde consumption and the accumulation of the reaction end products (Figure 6.1). Figure 6.2 illustrates kinetic trajectories of the reduced contributions of individual steps, computed by equations (6.1)- (6.6) for the same conditions... 

The results obtained validate the conclusion made on the basis of the analytical solution. Actually, it follows from Figmes 6.1 and 6.2 that at detectable conversion of benzaldehyde (for t >10 s), steps (3) and (4) promote increasing in the reaction selectivity, while steps (5) and (6) act inversely. In this case there is no necessity to introduce the approximation ri r4 to solve the problem. It follows from the kinetic trajectories of contributions for steps (1) and (4) that the higher is the aldehyde conversion the lower is the role of step (1) in free radical... 

Figure 6.2. Kinetic trajectories of the reduced value contributions of individual steps for liquid-phase oxidation of benzaldehyde. The curves are numbered in accordance with the steps.

Information forming the basis of the kinetic trajectories of the value contributions for steps involving an inhibitor and its transformation products may be useful for the selection of the molecular stmcture of the efficient inhibitor resulting in the maximum deceleration of a reaction. However this task may be solved strictly by finding the optumum inhibitor stmcture and the reaction conditions [31,32]. 

Kinetic trajectories of the value contributions of individual steps. Computing was carried out for the following initial conditions [RH]o=7.82 and 7.35 M, at 7 =60 and 120 °C, respecttively, [ROOHJo = 10 M, [O2] = 10 M = const. From kinetic trajectories of the value contributions of individual steps (Figme 7.6a,b) in the induction period, the following specific features of ethylbenzene oxidation inhibited by ara-substituted phenols may be marked out. 

Figure 7.6. Kinetic trajectories of reduced value contributions of steps over the induction period for liquid-phase oxidation of ethylbenzene inhibited by ora-methylphenol at 60 °C (a) and 120 °C (b). Initial concentrations of /lora-methylphenol and ethylbenzene hydroperoxide were lO M and 10 M, respectively (curves are numbered in accordance with step numbers in Table 7.1, the arrow points out the end of the induction period).

The characteristic time intervals mentioned above, selected from the kinetic trajectories of the value contributions, correlate with the kinetics of phenoxyl radical accumulation. It follows fiom Figure 7.8 drat completion of the first time interval (ti) corresponds to establishing die quasi-stationary mode of pora-methylphenoxyl radical accumulation. Over the time interval t2 - h the growth in the concentration of phenoxyl radicals is observed as a consequence of increasing the role of degenerate chain branching steps. Finally, mounting to the maximum value, the concentration of phenoxyl radicals decreases because of inhibitor consumption. 

As is the cases in earlier chapters, the function in (4.17) is zero at stationary states, increases on removal from stable stationary states and decreases from any initial given state on its approach to the nearest stable stationary state along a deterministic kinetic trajectory. These specifications make a Liapunov function in the vicinity of stable stationary states, which indicates the direction of the deterministic motion. Hence for every variation from a stable stationary state we have... 

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