Stationary/steady state approximation

This procedure constitutes an application of the steady-state approximation [also called the quasi-steady-state approximation, the Bodenstein approximation, or the stationary-state hypothesis]. It is a powerful method for the simplification of complicated rate equations, but because it is an approximation, it is not always valid. Sometimes the inapplicability of the steady-state approximation is easily detected for example, Eq. (3-143) predicts simple first-order behavior, and significant deviation from this behavior is evidence that the approximation cannot be applied. In more complex systems the validity of the steady-state approximation may be difficult to assess. Because it is an approximation in wide use, much critical attention has been directed to the steady-state hypothesis. 

Assuming that the catalytic reaction takes place in a flow reactor under stationary conditions, we may use the steady state approximation to eliminate the fraction of adsorbed intermediate from the rate expressions to yield ... 

Equations (5.16) of Table 5.1 refer to series first-order reactions. Of interest for the solvent extraction kinetics is a special case arising when the concentration of the intermediate, [Y], may be considered essentially constant (i.e., d[Y]/dt = 0). This approximation, called the stationary state or steady-state approximation, is particularly good when the intermediate is very reactive and present at very small concentrations. This situation is often met when the intermediate [Y] is an interfacially adsorbed species. One then obtains... 

The following treatment applies to the case where the solids are stationary in a shallow packed bed, so that they can be considered to be in well-mixed conditions, and that the solute initially saturates the solid, as in the case of vegetable oil in crushed seeds. For the quasi--steady-state approximation, Brunner [51] derived a practical equation ... 

A. Quasi-Stationary and Quasi-Steady-State Approximations. 105... 

The absorption cross sections for NO2 and the corresponding quantum yields are given in Table 8 and 9, respectively. The photolysis of NO2 has been investigated intensively over the last 40 years because of its critical role in the formation of ozone in the polluted tropospheric boundary layer [56-63]. The three reactions of Eqs. 33 and 34 form the basis for the photochemical production of ozone. If one considers only these three reactions, then the photo-stationary state (or photochemical steady-state approximation) can be invoked around the oxygen atom as follows ... 

To solve this differential equation in conjunction with a similar one for species A would be very difficult, and recourse is usually made to the steady-state approximation. This assumes that dC. /dt 0, or that the right-hand side of Eq. d is in a pseudo-equilibrium or stationary state. Justification for this was provided in the last example. 

The rigorous combination of these three consecutive rate steps leads to a very complicated expression, but this needs to be done only in principle for transient conditions, although even then a sort of steady-state approximation is often used for the surface intermediates in that it is assumed that conditions on the surface are stationary. The rates of change of the various species are... 

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. 

The term steady state is also used to describe a situation where some, but not all, of the state variables of a system are constant. For such a steady state to develop, the system does not have to be a flow system. Therefore, such a steady state can develop in a closed system where a series of chemical reactions take place. Literature on chemical kinetics usually refers to this case, calling it steady-state approximation. Steady-state approximation, occasionally called stationary-state approximation, involves setting the rate of change of a reaction intermediate in a... 

Reaction mechanisms may be developed to explain rate laws, but often they are developed in parallel, one helping the other. In translating a mechanism into a rate law a useful tool is Bodenstein s steady state approximation (SSA) or stationary state hypothesis. This approximation assumes that after a very short interval of time any reactive intermediate, which because of its reactivity will only be present in negligible proportions, will have its rate of decay equal to its rate of production, i.e., it will reach a steady concentration on a vanishingly small time scale. If this did not happen the amount of the intermediate would build up to measureable proportions and it would become an intermediate product. It is assumed at any instant that dcj /dt = 0, where R is the reactive intermediate. 

This equation is analogous to Eq. (5.4.18) or (5.4.19) for the steady-state current density, although the instantaneous current depends on time. Thus, the results for a stationary polarization curve (Eqs (5.4.18) to (5.4.32)) can also be used as a satisfactory approximation even for electrolysis with the dropping mercury electrode, where the mean current must be considered... 

For the analysis of nonlinear cycles the new concept of kinetic polynomial was developed (Lazman and Yablonskii, 1991 Yablonskii et al., 1982). It was proven that the stationary state of the single-route reaction mechanism of catalytic reaction can be described by a single polynomial equation for the reaction rate. The roots of the kinetic polynomial are the values of the reaction rate in the steady state. For a system with limiting step the kinetic polynomial can be approximately solved and the reaction rate found in the form of a series in powers of the limiting-step constant (Lazman and Yablonskii, 1988). 

From the stationary state approximation, d[A ]/dt <= 0 and d[A ]/dt = 0, the expressions for the steady state concentration of [A ] and [A ] can be set up. We can further define, 6m, and as quantum efficiencies of emission from dilute solution, from concentrated solution with quenching, and of excimer emission, respectively ... 

Fig, 47. Steady-state rate coefficient for reaction between diffusing particles, of diffusion coefficient J5, and stationary sinks, of radius R and volume fraction c. The rate coefficients are scaled to the Smoluchowski value of 4irRD for dilute solutions, the approximate expression of Felderhof and Deutch [25] [eqn, (241)], — — — ,... 

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