Dynamics concentration

Statement 4. Results similar to the Statement 3 hold, if in a set of the kinetic equations (8.2.17), (8.2.22) and (8.2.23) concentrations are no longer fixed but the values of N = (3/K, N, = a/K corresponding to the solution of the concentration dynamics are used. The relevant marginal parameter is ( )/ ) This statement could be proved in the same way as above. Substitution of A a — 1 1 K, N = a/K changes slightly a form of the nonlinear kinetic equations (taking into account the K definition (8.2.14). All peculiarities of the d = 1,2 class remain valid. 

The performed calculations demonstrate that a type of the asymptotic solution of a complete set of the kinetic equations is independent of the initial particle concentrations, iVa(O) and A b(O)- Variation of parameters a and (3 does not also result in new asymptotical regimes but just modifies there boundaries (in t and k). In the calculations presented below the parameters A a(O) = Ab(0) = 0-1 and a = /3 = 0.1 were chosen. The basic parameters of the diffusion-controlled Lotka-Volterra model are space dimension d and the ratio of diffusion coefficients k. The basic results of the developed stochastic model were presented in [21, 25-27]. 

Let us consider a projection of the complex many-dimensional motion (which variables are both concentrations and the correlation functions) onto the phase plane A b) - It should be reminded that in its classical formu- 

Therefore, oscillations of K (t) result in the transition of the concentration motion from one stable trajectory into another, having also another oscillation period. That is, the concentration dynamics in the Lotka-Volterra model acts as a noise. Since along with the particular time dependence K — K t) related to the standing wave regime, it depends also effectively on the current concentrations (which introduces the damping into the concentration motion), the concentration passages from one trajectory onto another have the deterministic character. It results in the limited amplitudes of concentration oscillations. The phase portrait demonstrates existence of the distinctive range of the allowed periods of the concentration oscillations. 

For a given set of parameters the period of concentration oscillations (or its average for a periodic motion) exceeds greatly the period of the correlation motion. For the slow concentration motion not only the period of the standing wave oscillations but also their amplitudes and, consequently, the amplitude in the K (i) oscillations depend on the current concentrations N t) and N j t). In other words, the oscillations of the reaction rate are modulated by the concentration motion. Respectively, the influence of the time dependence K = K t) upon the concentration dynamics has irregular, aperiodic character. A noise component modulates the autowave component (the standing waves) but the latter, in its turn, due to back-coupling causes transition to new noise trajectories. What we get as a result is aperiodic motion chaos). The mutual influence of the concentration and correlation motions and vice versa is illustrated in Fig. 8.2, where time developments of both the concentrations and reaction rates are plotted. 

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Monomer concentration dynamics are presented in Figure 5. Additional observations for Run 5 are accurately correlated during the reactor startup and at final steady state. The observation at one residence time, Run 4, may be in error. The total cummu-lative, molar concentrations of macromolecules as a function of time are presented in Figure 6. The errors associated with this dependent variable are also evident during the steady state analysis of initiation... 

Figure 6. Cumulative molar concentration dynamics (--------) Run 4 simulation...

INITIAL specifies the start of the INITIAL region specify the initial concentrations DYNAMIC specifies the start of the DYNAMIC region represent the model equations is a check on the total mass balance... 

Observed monomer concentrations are presented by Figure 2 as a function of cure time and temperature (see Equation 20). At high monomer conversions, the data appear to approach an asymptote. As the extent of network development within the resin advances, the rate of reaction diminishes. Molecular diffusion of macromolecules, initially, and of monomeric molecules, ultimately, becomes severely restricted, resulting in diffusion-controlled reactions (20). The material ultimately becomes a glass. Monomer concentration dynamics are no longer exponential decays. The rate constants become time dependent. For the cure at 60°C, monomer concentration can be described by an exponential function. 

Leonard, C. S., Michaelis, E. K. Mitchell, K M. (2001). Activity-dependent nitric oxide concentration dynamics in the laterodorsal tegmental nucleus in vitro. 

Fig. 22. Concentration dynamics resulting from a step change of T0 from 573 to 593 Kandof xgo from 0.06 to 0.07, type II conditions.

It is convenient to divide a set of fluctuation-controlled kinetic equations into two basic components equations for time development of the order parameter n (concentration dynamics) and the complementary set of the partial differential equations for the joint correlation functions x(r, t) (correlation dynamics). Many-particle effects under study arise due to interplay of these two kinds of dynamics. It is important to note that equations for the concentration dynamics coincide formally with those known in the standard kinetics... 

Fig. 4.1. The schematic relation between concentration dynamics and correlation dynamics in...

The non-linearity of the equations (5.1.2) to (5.1.4) prevents us from the use of analytical methods for calculating the reaction rate. These equations reveal back-coupling of the correlation and concentration dynamics - Fig. 5.1. Unlike equation (4.1.23), the non-linear terms of equations (5.1.2) to (5.1.4) contain the current particle concentrations n (t), n t) due to which the reaction rate K(t) turns out to be concentration-dependent. (In particular, it depends also on initial reactant concentration.) As it is demonstrated below, in the fluctuation-controlled kinetics (treated in the framework of all joint densities) such fundamental steady-state characteristics of the linear theory as a recombination profile and a reaction rate as well as an effective reaction radius are no longer useful. The purpose of this fluctuation-controlled approach is to study the general trends and kinetics peculiarities rather than to calculate more precisely just mentioned actual parameters. 

After some mathematical manipulations with (5.2.3) we arrive at the equations of the concentration dynamics... 

A set of equations (8.2.12) and (8.2.13) for the concentration dynamics is formally similar to the standard statement of the Lotka-Volterra model given... 

Statement 2. Substitution into the concentration dynamics (equations (8.2.12) and (8.2.13)) of the reaction rate K — K(Na, Nb), dependent on the current concentrations, changes the nature of the singular point. In particular, a centre (neutral stability) could be replaced by stable or unstable focus. This conclusion comes easily from the topological analysis its illustrations are well-developed in biophysics (see, e.g., a book by Bazikin [30]). 

Attach the sampling tube to a purge-and-trap concentrator (dynamic headspace). 

The advent of nanosecond time-resolved fluorimetry in the late 1970s and early 1980s made it possible to obtain fluorescence lifetimes of the prototropic forms as a function of proton concentration. Dynamic analysis at nanosecond resolution enabled a more accurate estimation of pK a values. Phenomenon like pro-ton-induced fluorescence quenching in naphthylamines could then be given a satisfactory explanation. Shizuka s review [18] in 1985 summarizes the dynamic analysis techniques that were employed. Certain intermolecular proton-transfer... 

Ledo A, Frade J, Barbosa RM, Laranjinha J. Nitric oxide in brain diffusion, targets and concentration dynamics in hippocampal subregions. Mol Asp Med 2004 25 75. 

In this chapter, we present most of the equations that apply to the systems and processes to be dealt with later. Most of these are expressed as equations of concentration dynamics, that is, concentration of one or more solution species as a function of time, as well as other variables, in the form of differential equations. Fundamentally, these are transport (diffusion-, convection-and migration-) equations but may be complicated by chemical processes occurring heterogeneously (i.e. at the electrode surface - electrochemical reaction) or homogeneously (in the solution bulk chemical reaction). The transport components are all included in the general Nernst-Planck equation (see also Bard and Faulkner 2001) for the flux Jj of species j... 

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