Steady-state reaction rates

Thus, this eigenvalue detenuines the unimolecular steady-state reaction rate constant. 

Figure 2 shows the effect of ethylene pressure upon the steady-state reaction rates for clean Ag(llO) at 490 K and 150 torr O2 (25). Also shown is the steady-state coverage of atomically adsorbed oxygen... 

Figure 3. The effect of O2 steady state reaction rates at 490 K and =4.1 torr...

Finally, we note the differences between a Ru(OOOl) catalyst with or without added Cu with respect to attaining steady-state reaction rates. On the Cu-free... 

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 catalytic cycle without limitation can have relaxation time much bigger then 1 /fcmin/ where fc in is the minimal reaction rate constant. For example, if all k are equal, then for m = 11 we get xx20/k. In more detail the possible relations between t and the slowest constant were discussed by Yablonskii and Cheresiz (1984). In that paper, a variety of cases with different relationships between the steady-state reaction rate and relaxation was presented. 

The steady-state reaction rate and relaxation time are determined by these two constants. In that case their effects are coupled. For the steady state we get in first-order approximation instead of Equation (13) ... 

In all four cases, tlie initial reaction rates at the start of illumination in the continuous-feed photoreactor were higher than the pseudo-steady-state reaction rates the reaction rates declined over time until pseudo-steady-state operation was achieved. Tliis apparent deactivation phenomenon, often observed with aromatic contaminants, is discussed in Sec. III.E. In a transient reaction system, the time required to reach pseudo-steady-state operation also appears to increase in the same order as the reaction rates. For example, for the continuous photocatalytic oxidation of aromatic contaminants at 50 mg/m in a powder-layer photoreactor, the time required for pseudo-steady-state operation to be achieved was reported to be approximately 90 min for benzene, 120 min for toluene, and as long as 6 hr for wz-xylene [50,51]. Under such conditions, the difference in reaction rates between the aromatic contaminants is magnified by the fact that the more reactive aromatics retain their higher transient reaction rates for longer periods (Fig. 7). 

As it was said above (Section 3.2), for the elastic interaction this coefficient coincides with the effective radius of recombination, Reff = b, whereas for the Coulomb interaction Re ff is defined in equation (3.2.51). Therefore the problem of obtaining the steady-state reaction rate is reduced to the finding the asymptotic coefficient b of the solution of equation (4.2.25). Formally it coincides with the quantum-mechanical scattering length on the potential... 

Since equation (4.3.11) is quite similar to the Fermi-Dirac distribution, we can approximate it by step-like function which is zero as r Rn and unity if r > f H- When doing so, we obtain from equation (4.3.10) the steady-state reaction rate... 

FIGURE 3 Three-dimensional view of the steady-state reaction rate surface when -y, = 0.001, and y-i = 0.002. The inset shows a isola for a section of constant at where H and H are Hopf bifurcations and T and T are turning points. 

FIGURE 4 Relation between Figures 3,5 and 8. The steady-state reaction-rate surface has bifurcation curves scribed on its surface which can be projected onto the parameter plane below or along lines of constant a to give side view. 

FIGURE 5 Side views of the steady-state reaction rate surface for y, = 0.001 and y2 = 0.002. (a) Left-hand or S > 0 side of the surface (b) right-hand or 8 < 0 side of the surface. Points K, L, M and N correspond to double zero eigenvalues and Q is the projection of the state labelled the same in Figures I and 9. 

FIGURE 6 Demonstration of the relation between the function and the shape of the phase plane, (a) Phase plane contours of the function d> = (I - 0 - ft) = A. (b) Phase portrait for the special case of negligible desorption and identical absorption rates (< i = a2 = A -y, = y2 = 0). (c) Shape of the non-unique steady-state reaction rates for variations in reactant partial pressures for the case of negligible desorption. 

There was always an induction period of 10 to 20 min before the benzene product reached its steady-state rate of production as detected by the mass spectrometer after the introduction of cyclohexane onto the crystal surface. This is shown in Fig. 22 for several catalyst temperatures. The catalyst was initially at 300 K. When steady-state reaction rates were obtained, the catalyst temperature was rapidly increased (in approximately 30 sec) to 423 K and the reaction rate monitored. This was repeated with heating to 573 and 723 K. The benzene desorbed during rapid heating of the catalyst surface is approximately 1 x 1013 molecules or less and represents only a small fraction of the carbon on the surface. The steady-state reaction rates at a given temperature are the same whether the catalyst was initially at that temperature or another. This induction period coincides with a higher than steady-state uptake of cyclohexane. A mass balance calculation on carbon, utilizing the known... 

Finally, the steady-state reaction rate ks is calculated as the ratio of the current J0 to the reactant population N(xR), that is,... 

With fixed [CO], and increasing [02] the steady-state reaction rate W is initially zero (the overall surface is covered with CO) and it can then jump to the value k22[CO]212kt [02]. With further increase in [02], the reaction rate varies inversely with [02]. In turn, with constant [02] the reaction rate rises quadratically with increasing [CO] and then "jumps down to zero values. This example indicates that rather simple but non-linear schemes can be characterized by complex dynamic behaviour. In radiophysical terms, scheme (119) can be called a simple catalytic trigger since, in this case, there exist two stable steady states. 

It is well-known that the difference of parameter values results in the indeterminacy of parameters. Rate limitation and the steady-state reaction rate will be dependent only on the parameters of "slow steps. But this case is beyond the scope of our discussion here. 

Here there always exist two boundary steady states (x = 1, y = 0) and (x = 0, y = 1). The former corresponds to the complete surface coverage with substance A2 and the latter by substance B. In both cases the steady-state reaction rate W = k3xy = 0. But besides boundary steady states, there can also exist internal steady states. After subtracting eqn. (10b) from eqn. (10a), we obtain... 

We can write down relationships to determine critical values for the parameters at which a transition from one branch of the curve for the dependence of the steady-state reaction rate to the other becomes possible. For example, at fixed 3 and PA, the critical value of PB is determined by the equation... 

Fig. 1. Dependence of steady-state reaction rate on partial pressures PAi. (a) and PB (b). 1, (non-zero branch) 2, VVltls (zero branch) 3, Wlm Act - critical point of drop in steady-state reaction rate B = limit value of Ai r.

Fig. 7. Steady-state reaction rate surface at T = const.

In several experiments, in particular the study by Temkin and co-workers [224] of the kinetics in ethylene oxidation, slow relaxations, i.e. the extremely slow achievement of a steady-state reaction rate, were found. As a rule, the existence of such slow relaxations is ascribed to some "side reasons rather than to the purely kinetic ("proper ) factors. The terms "proper and "side were first introduced by Temkin [225], As usual, we classify as slow "side processes variations in the chemical or phase composition of the surface under the effect of reaction media, catalyst deactivation, substance diffusion into its bulk, etc. These processes are usually considered to require significantly longer times to achieve a steady state compared with those characterizing the performance of chemical reactions. The above numerical experiment, however, shows that, when the system parameters attain their bifurcation values, the time to achieve a steady state, tr, sharply increases. 

Fig. 17. Steady-state reaction rate as a function of (a) k1 and (b) k2. The broken lines are unstable steady states.

Fig. 15. Time to achieve steady-state reaction rate values as a function of gradual stepwise temperature variation.

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