Rate quasi-stationary state

In treating parallel reaction, two concepts are often used (i) the concept of rate-determining path, in which the fastest path is the rate-determining path, and (ii) the concept of steady state, also called the concept of quasi-stationary states of trace-level intermediates. 

The increase in the rate of oxidation of methanol with supercritical water concentration at 500 °C has been attributed to the increase in the concentration of OH radicals in a quasi-stationary-state.285... 

From the reaction mechanism, we can derive a reaction rate equation (kinetics) for the overall reaction. Below, we will discuss two such methods rate limiting step and quasi-stationary state (pseudo-steady-state). 

Quasi-Stationary State. The assumption in this method is that the concentration of intermediates is constant, after a short initial period. This means that the rate of formation of intermediates is equal to the rate of their disappearance/consumption. 

The mathematical techniques most commonly used in chemical kinetics since their formulation by Bodenstein in the 1920s have been the quasi-stationary state approximation (QSSA) and related approximations, such as the long chain approximation. Formally, the QSSA consists of considering that the algebraic rate of formation of any very reactive intermediate, such as a free radical, is equal to zero. For example, the characteristic equations of an isothermal, constant volume, batch reactor are written (see Sect. 3.2) as... 

The Bodenstein approximation is accurate within reason, provided the intermediate is and remains at trace level, and with the exception of a very short initial time period in which the quasi-stationary state is established [13-15], It is left to the practitioner to decree how low a concentration must be to qualify as "trace " the more generous he is, the less accurate will be his results. For the pathway 4.20 with one trace-level intermediate, the error introduced can be estimated in the same way as for rate control by a slow step (see Section 4.1.1) ... 

The three principal tools for reduction of mathematical complexity of rate equations are the concepts of a rate-controlling step, of quasi-equilibrium steps, and of quasi-stationary states of trace-level intermediates. 

Any highly reactive intermediate that is and remains at trace level attains a quasi-stationary state in which its net chemical rate is negligibly small compared separately with its formation and decay rates. This is the basis of the Bodenstein approximation, which allows the rate equation of the intermediate to be replaced by an algebraic equation for the concentration of the intermediate, an equation which can then be used to eliminate that concentration from the set of equations. The approximation can be applied in succession for each trace-level intermediate. It is the most powerful tool for reduction of complexity. It is the basis of general formulas to be introduced in Chapter 6 and widely used in subsequent chapters. 

A general formula for single catalytic cycles with arbitrary number of members and arbitrary distribution of catalyst material has been derived by Christiansen. Unfortunately, the denominator of his rate equation for a cycle with k members contains k2 additive terms. Such a profusion makes it imperative to reduce complexity. If warranted, this can be done with the concept of relative abundance of catalyst-containing species or the approximations of a rate-controlling step, quasi-equilibrium steps, or irreversible steps, or combinations of these (the Bodenstein approximation of quasi-stationary states is already implicit in Christiansen s mathematics). In some fortunate instances, the rate equation reduces to a simple power law. 

Instead of the quasi-stationary state assumption of Kramers, he assumed only that the density of particles in the vicinity of the top of the barrier was essentially constant. Visscher included in the Foldcer-Planck equation a source term to accoimt for the injection of particles so as to compensate those escaping and evaluated the rate constant in the extreme low-friction limit. Blomberg considered a symmetric, piecewise parabolic bistable potratial and obtained a partial solution of the Fokker-Hanck equation in terms of tabulated functions by requiring this piecewise analytical solution to be continuous, the rate constant is obtained. The result differs from that of Kramers only when the potential has a sharp, nonharmonic barrier. 

Assuming the existence of a quasi-stationary state the rate constant of an exothermic electron-transfer reaction can be written as... 

A block diagram of the lamp circuit is shown in Fig. 38. Each pulse produced 10" -10" % decomposition. They were fired at the rate of 30-40 pulses per second, sufficient to produce a quasi-stationary state. The radical concentration following each pulse falls to the same value [J ]d prior to the pulse (Fig. 39). A consideration of the rate of formation of CjHg and CH leads to an expression from which and ki5 may be determined by varying the length of the dark period t. Paired pulses with a varying t should lead to direct evidence for the participation of hot radical reactions. 

Implementing the added mass force has barely any influence on the steady state solution [30, 66]. Been et al [30] explained this to some extent surprising result by the fact that the simulations soon reach a quasi-stationary state where there is only minor acceleration. The bubble jets observed close to the distributor plate are then disregarded. However, the convergence rate and thus the computational costs are often significantly improved implementing this force. 

In the interruption test (Kressman and Kitchener, 1949), the ion exchanger is temporarily separated from the liquid. If the rate is controlled by mass transfer in the liquid, the rate upon reimmersion is the same as at the time of separation, the quasi-stationary state in the film is very quickly re-established. If the rate is controlled by intraparticle diffusion, concentration gradients within the particle have time to relax, and the rate is faster upon reimmersion. This comparison is independent of specific mechanisms and algebraic forms of rate laws. 

Chemical reactions may involve large numbers of steps and participants and thus many simultaneous rate equations, all with their temperature-dependent coefficients. The full set of rate equations is easily compiled as shown in Section 2.4, and to obtain solutions by numerical computation poses no serious problems. With a large number of equations, however, it may become too much of a task to verify the proposed network and obtain values for all its coefficients. Therefore, every available tool must be brought to bear to reduce the bulk of mathematics, and that without unacceptable sacrifice in accuracy. The present chapter critically reviews the principal tools for such a purpose stoichiometric constraints and the concepts of a rate-controlling step, quasi-equilibrium steps, and quasi-stationary states. Other tools useful in catalysis, chain reactions, and polymerization will be discussed in the context of those reactions (see Sections 8.5.1, 9.3, 10.3, and 11.4.1). 

The net rate through each of the steps of a pathway at quasi-stationary state is practically the same. 

Any highly reactive intermediate that is and remains at trace level attains a quasi-stationary state in which its net chemical rate is negligible compared separately with its formation and decay rates. This is the basis of the Bodenstein approximation, which... 

A quasi-stationary state is stable if a small excursion from it is self-correcting, but is unstable if the excursion escalates. Specifically, in a system as described here, stability is ensured if the net rates rx and rY decrease if the concentrations of X and Y increase. If this is true for one of the intermediates, but not for the other, the stabilizing and destabilizing tendencies counteract one another, and the (necessary and sufficient) condition for stability becomes... 

Example 14.4. Test for stability of quasi-stationary states [35], Let the (multistep) partial reaction X— Y and its rate equation be... 

Third, it is often useful to assume that the concentration of one or more of the intermediate species is not changing very rapidly with time (i.e., that one has a quasi-stationary-state situation). This approximation is also known as the Bodenstein steady-state approximation for intermediates. It implies that the rates of production and consumption of intermediate species are nearly equal. This approximation is particularly good when the intermediates are highly reactive. 

This ratio is readily obtained from the quasi-stationary state approximation, noting that V = v = V2- The use of Temkin relation illustrates how the unknowns can be eliminated by alternation of the indices. Thus the rate can always be obtained, at least in principle, although it can be difficult when the rates of elementary steps are not linear with the concentration of active intermediates, e.g., for dissociated adsorption.The turnover rate is then ... 

The quasi-stationary state approximation leads to a rate expression ... 

This important equation, first derived by Kramers (1940), allows an easy derivation of the rate constant defined as the ratio 0/nj, diere n is the number of psurticles A. A harmonic potential is taken for U near A = 1/2C0aX ) quasi stationary state, w is taken aa zero... 

All dependences considered above were obtained under the assumption that the quasi-stationary concentration of RO-2 is established in the system very rapidly (during the time of heating of the reactor). This is valid only at such a sufficiently hi Vfo at which the time of establishment of the quasi-stationary state Xj, = 0.74( i7V,o) is sh( ter than the time of heating. Since the rates of chain initiation during the autoxidation of RH are often very low, Xj, can be raflier noticeable value. For example, at v,o = 10 l/(mol s) and Ar = 10 l/(mol s) Xj, = 750 s and the time of heating usually does not exceed 150 s. In these cases, formula (11.20) gains somewhat different form... 

---