Reaction rate K t

Equation (6.2.15) with a = d/A corresponds to the reaction rate whose time decay is characterized by the reaction rate K(t) oc 3. The 

As it is noted in Section 5.1, a distinctive feature of the linear approximation is the absence of back-coupling between the concentration n t) and the correlation function Y (r, t) which is also independent of the initial reactant concentrations. Moreover, in the linear approximation the parameter K = D /D does not play any role at all. So, in the black-sphere model for the standard random distribution, Y r ro,t) = 1, one gets universal relations (4.1.69), (4.1.65) and (4.1.61) for d = 1,2 and 3 respectively. In contrast, in the superposition approximation the law K — K t) loses its universality, since along with space dimension d it depends also on both the parameter k and the initial reactant concentrations. 

When the initial concentrations are small, fluctuation effects are not pronounced and the curve of the superposition approximation degenerates to those of the linear approximation. However, for any initial concentration no there exists distinctive time to — to no), at which these approximations start to diverge from each other. For d = 2 parameter k slightly affects behaviour of the K = K (f) - full and broken curves 1 at long times t reveal the same slopes with the only difference in co-factors. At last, for d = 3 curves at K = 0 and K = 0.5 differ considerably and have different slopes at long t (this peculiarity of the kinetics when one of the reactants is immobile has been already stressed above). 

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N E) is called the cumulative reaction probability. It is directly related to the themial reaction rate k T) by... 

A new principal element of the Waite-Leibfried theory compared to the Smoluchowski approach is the relation between the effective reaction rate K(t) and the intermediate order parameter x = Xab (r,i). In its turn, the Smoluchowski approach is just an heuristic attempt to describe the simplest irreversible bimolecular reactions A + B- B,A + B- B and A + B -> 0 and cannot be extended for more complicated reactions. The Waite-Leibfried approach is not limited by these simple reactions only it could be applied to the reversible reactions and reaction chains. However, in the latter case the particular linearity in the joint correlation function x = Xab (r, ) does not always mean linearity of equations since additional non-linearity caused by particle densities can arise. 

What was said above is illustrated by Fig. 1.29 and Fig. 1.30 corresponding to the cases p > pc and p < pc respectively. To make the presented kinetic curves smooth, in these calculations the transformation rate A — B was taken to be finite. To make results physically more transparent, the effective reaction rate K (t) of the A —> B transformation is also drawn. The standard chemical kinetics would be valid, if the value of K (t) tends to some constant. However, as it is shown in Fig. 1.30, K(t) reveals its own and quite complicated time development namely its oscillations cause the fluctuations in particle densities. The problems of kinetic phase transitions are discussed in detail in the last Chapter of the book. 

Therefore, if the reaction rate K (t) is known, the A+B -4 0 reaction kinetics is defined uniquely. 

Transient kinetics with the time-dependent reaction rate K(t), equation (4.1.61), have been observed experimentally more than once (e.g., by Tanimura and Itoh [19]). In the steady-state case, t — oo, we get finally... 

In order to analyze the temporal process of steady-state formation in a course of tunnelling reactions in crystals, Kotomin [85] solved numerically equation (4.3.28) and the relevant equation (4.1.19) for the reaction rate K(t). It is clearly seen in Fig. 4.11 that the steady-states formed after the transient... 

Figure 4.21 shows the diffusion kinetics (equations (4.1.18), (4.1.19) and (4.1.23)) of an initial stage of the thermal stimulation made after a static tunnelling decay for a certain time in all cases one can see the luminescence increase. It is greater, the closer the luminescence intensity at the instant stimulation moment to its quasi-steady value (broken line). Calculations were carried out assuming that the temperature stimulation yields relatively weak perturbation, i.e., the defect concentration n(t) in equation (4.1.18) changes much more slowly than the reaction rate K(t) and therefore I(t) oc K (t). 

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. 

Despite the fact that (5.3.3) reveals the same asymptotic behaviour (nA(t) oc f-1/2, as t —> oo), the relative concentration is always smaller than predicted by the Smoluchowski theory for nA — l/2nA(0), the discrepancy is 9%. In other words, the Smoluchowski approach slightly underestimates a real reaction rate due to its neglect of reactant density fluctuations stimulating. (Note that in the case of second reaction, A + A — A, the reaction rate K(t) in the Smoluchowski approach has to be corrected by a factor 1/2... 

Fig. 5.18. The non-stationary part of the reaction rate K(t) for the transient kinetics of the A + A 0 reaction (curves 1 to 3) and A + B —> 0 reaction (curve 4), d = 3 (random initial distribution) [101]. The broken line shows prediction of the standard chemical kinetics, equation (5.3.10). The initial concentrations are given.

If the exponent a turns out to be less than its classical value found in the linear approximation, ao = 1 (d 2), the relevant reaction rate K(t) is reducing at long times so that in the long-time limit K(oo) = 0 (the reaction rate s zerofication). On the other hand, when searching the asymptotic solution of non-linear equations, the asymptotic relation (6.2.4) permits to replace tK(t)n(t) for a. 

The kinetics of the A + B - 0 bimolecular reaction between charged particles (reactants) is treated traditionally in terms of the law of mass action, Section 2.2. In the transient period the reaction rate K(t) depends on the initial particle distribution, but as f -> oo, it reaches the steady-state limit K(oo) = K() = 47rD/ieff, where D — Da + >b is a sum of diffusion coefficients, and /4fr is an effective reaction radius. In terms of the black sphere approximation (when AB pairs approaching to within certain critical distance ro instantly recombine) this radius is [74]... 

The equation for the time development of macroscopic concentrations formally coincides with the law of mass action but with dimensionless reaction rate K(t) = K(t)/ AnDr ) which is, generally speaking, time-dependent and defined by the flux of the dissimilar particles via the recombination sphere of the radius tq, equation (5.1.51). Using dimensionless units n(t) = 4nrln(t), r = t/tq, t = Dt/r, and the condition of the reflection of similar particles upon collisions, equation (5.1.40) (zero flux through origin), we obtain for the joint correlation functions the equations (6.3.2), (6.3.3). Note that we use the dimensionless diffusion coefficients, a = 2k, IDb = 2(1 — k), k = Da/ Da + Dq) entering equation (6.3.2). 

To demonstrate the charge screening importance in a many-particle system, it should be mentioned that the substitution of potentials il (r, t) entering equations (6.3.2), (6.3.3) by non-screened potentials Uu(r) = L/r leads to the Coulomb catastrophe manifesting itself by the unlimited increase of the reaction rate K(t) [24, 78],... 

Fig. 6.41. The time-development of the reaction rate K(t). Full curve - symmetric case (Da = Db), broken curve - asymmetric case (DA = 0), dotted curve - D /D = 0.01.

In its turn, the non-monotonous behaviour of Y(r, t) results in a similar behaviour of the reaction rate K(t) in time (Fig. 6.48). The local maximum of K (f) observed at t = 101 (for a given L) for different initial concentrations likely arises due to the initial conditions used, which do not take into account peculiarities of the spatial distribution of charged particles a more adequate one would be a quasi-equilibrium pair distribution with incorporated potential screening. 

The conclusion could be drawn from Fig. 8.2 that the peaks in K(t) produce a fine structure in the concentration curves. Despite the fact that these oscillations in K(t) have two orders of magnitude, the fine structure is not of a primary importance. In its turn, the concentration oscillations modulate oscillations of the reaction rate K(t). 

These curves show four kinds of structures which are dependent on the current particle concentrations and the oscillation phases of the reaction rate K(t). The moment of time t = 295.0 corresponds to the K(t) maximum whose concentration Na(t) is close to its minimum value. The behaviour of the correlation functions reminds that shown in Fig. 8.5 but the function for the dissimilar particles has now maximum. After a short time interval, at t = 296.0, despite very small change of concentrations and the correlation functions for similar particles, the maximum in the correlation functions for dissimilar particles completely disappeared (K(t) has a minimum). 

This statement comes from analytical and topological studies [4], Unlike the Lotka-Volterra model where due to the dependence of the reaction rate K(t) on concentrations NA and NB, the nature of the critical point varied, in the Lotka model the concentration motion is always decaying. Autowave regimes in the Lotka model can arise under quite rigid conditions. It is easy to show that not any time dependence of K(t) emerging due to the correlation motion is able to lead to the principally new results. For example, the reaction rate of the A + B -> 0 reaction considered in Chapter 6 was also time dependent, K(t) oc t1 d/4 but its monotonous change accompanied by a strong decay in the concentration motion has resulted only in a monotonous variation of the quasi-steady solutions of (8.3.20) and (8.3.21) jVa(t) (3/K(t) and N, (t) p/f3 = const. 

Deviation from standard chemical kinetics described in (Section 2.1.1) can happen only if the reaction rate K (t) reveals its own non-monotonous time dependence. Since K(t) is a functional of the correlation functions, it means that these functions have to possess their own motion, practically independent on the time development of concentrations. The correlation functions characterize the intermediate order in the particle distribution in a spatially-homogeneous system. Change of such an intermediate order could be interpreted as a series of structural transitions. 

In a system with strong damping of the concentration motion the concentration oscillations are constrained they follow oscillations in the correlation motion. As compared to the Lotka-Volterra model, where the concentration motion defines essentially the autowave phenomena, in the Lotka model it is less important being the result of the correlation motion. This is why when plotting the results obtained, we focus our main attention on the correlation motion in particular, we discuss in detail oscillations in the reaction rate K(t). 

Figure 8.7 shows the dependence of the reaction rate K(t) for different k values. For the space dimension d = 3 the obtained results could be easily interpreted as follows there exists the marginal value of kq (Statement 2). For k = 0.05 the inequality k > o holds. The stable stationary solution exists for the correlation dynamics and due to decay of the concentration motion... 

Statement 1) a stable stationary solution of a complete set of equations of the Lotka model holds. At long t the reaction rate K(t) strives for the stationary value. Time development of concentrations obeys standard chemical kinetics, Section 2.1.1. 

Fig. 8.10. The Fourier spectrum of the reaction rate K t). Parameters k = 0.005, d = 3. Enumeration of curves (a) to (e) corresponds to the time intervals presented in Fig. 8.9.

The overall reaction rate has a temperature dependence governed by the specific reaction rate k(T) and a concentration dependence that is expressed in terms of several concentration-based properties depending on the suitability for the particular reaction type mole or mass concentration, component vapor partial pressure, component activity, and mole or mass fraction. For example, if the dependence is expressed in terms of molar concentrations for components A(Ca) and B(Cb), the overall reaction rate can be written as... 

The temperature-dependent specific reaction rate k(T) is represented by the Arrhenius equation... 

The modified rate equation (MRE) approximation [177] was also constructed to describe the non-Markovian character of diffusion-controlled reversible reactions. The forward reaction rate k(t) is the same as in contact DET, Eq. (3.21), but with ka substituted for ko. As for the backward reaction rate, it was modified to be proportional to k(t) ... 

Fig. 12 Arrhenius plot of the reaction rate k(T) measured in the presence of PS-PNIPA-Ag composite particles at different temperatures squares KPSl-Ag (2.5 mol% BIS), triangles KPS2-Ag (5 mol% BIS), and circles KPS3-Ag (10 mol% BIS). The concentrations of the reactants are composite particles, S = 0.042 m2 L-1 [4-nitrophenol] = O.lmmolL-1 [NaBIE] = lOmmolL-1. The...

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