The extended steady-state approximation

When the steady-state approximation of Section B.2.5.2 is applied to nonisothermal systems involving molecular transport, it is sometimes referred to as the extended steady-state approximation. The simplifications that have just been indicated to follow from the application of this approximation underscore the importance of having methods for ascertaining the validity of the approximation. Criteria for the applicability of the extended steady-state approximation have been developed by Giddings and Hirsch-felder [64] and improved by Millan and Da Riva [89]. The discussion 

A statement of the extended steady-state approximation is that throughout the flame, the net mass rate of production of any reaction intermediary r is negligibly small. 

The integral of equation (80) for i = r is then 6 = 0 because the initial (as well as final) chain-carrier concentration is zero. Since the production terms are functions of t and of the mole fractions relations such as equation (85) permit explicit representation of in terms of t and the remaining X., Thus if equation (85) is valid for all chain carriers, the mole fractions of all these reaction intermediaries may be eliminated from the flame equations, and the flux fractions of all of these species are zero. Since stoichiometry conditions relate the remaining 6 , only one independent expression remains in equation (80), and the problem is reduced to that of a one-step reaction. The flame equations may then be solved explicitly to give all the mole fractions, including the X, in terms of t. 

For the sake of simplicity, consideration will be restricted to systems in which only one intermediary exists this will be denoted by the subscript r, and the other species will be denoted by the subscripts j(j = 1. iV — 1). The discussion and results are applicable qualitatively for a given chain carrier in flames containing any number of chain carriers. 

Since reaction intermediaries are present in low concentrations, equation (81) may be simplified for species r. It can be shown [91] that when X 1, if the diffusion velocities are the same for all species other than r, then equation (81) reduces to 

---

A criterion for the appUcability of the extended steady-state approximation... 

When a preliminary extended version of the kinetic model is used to identify the B-Zh reaction mechanism, frequently classical approximation methods are used, such as the ratedetermining step and the quasi-steady state approximations (see, for example [22,24-26]). However T. Turanyi and S. Vajda [29], applying the sensitivity analysis method, more precisely the method of the principal components analysis selection, specified numerically a base mechanism that comprise only 9 steps for the B-Zh reaction (see table 8.1), from the conventional model of Edelson-Field-Noes (EFN), which includes 32 steps. In this case, the... 

Borghans, J.A.M., De Boer, R.J., Segel, L.A. Extending the quasi-steady state approximation by changing variables. Bull. Math. Biol. 58, 43-63 (1996)... 

Wegner and Engel (1975) extended the Oosawa analysis by dropping the assumption of irreversibility in polymerization, and these workers applied a steady-state approximation. Their approach focuses on a dimeric nucleus, and there is no easy way to extend their theory to much more cooperative systems and still obtain tractable rate expressions. 

The shaded portion of the top graph is shown in expanded form in the bottom graph. After a brief initial period (usually less than a few seconds) the concentration of ES remains approximately constant for an extended period. The steady-state approximation is applicable during this second period. Most measurements of enzyme kinetics are made in the steady state. 

The series model can be extended to longer series and to the inclusion of reversibility to illustrate a variety of fundamental kinetic phenomena in an especially simple and straightforward manner. Depending on the relative rates employed, one can demonstrate the classic kinetic phenomena of a rate-limiting step and preequilibrium,72 and one can examine the conditions needed for the validity of the steady-state approximation commonly used in chemical kinetics.70... 

The use of the steady-state approximation is justified on the basis of two separate, independent observations. Firstly, after sufficient polymer has accumulated, the rate remains constant over an extended range of conversion (Figure 4). Secondly, Bengoughs measurements (10) of the non-steady-state kinetics of acrylonitrile polymerization show that a steady state is established within minutes, whereas the polymerization continues for hours. 

To analyze the rate terms on the right-hand side of equation (2.6), the use of the partial equilibrium approximation is extended to permit evaluation of other unmeasured species concentrations which may enter these terms. This use of the measured main reaction progress to evaluate the concentration of a kinetically significant intermediate is closely analogous to the conventional quasi-steady state approximation in kinetics, but free of the usual restriction on its accuracy or utility that the concentration so evaluated be stoichiometrically minor. 

While we are still self-constrained to limit our treatment to what we believe is essential to physical chemistry, we have added further examples to the Chapter 7 treatment of reaction kinetics, which include some aspects of multistep mechanisms and introduced the steady-state approximation. The steady-state concept was then extended to the Eyring transition-state concept and used again for the critical step in the Michaelis-Menten treatment of enzyme kinetics. This has been a fast tour of some complicated algebra but in our experience students who learn the derivations have a deeper appreciation for the concepts. Casual interviews of students from past classes have revealed that the Michaehs-Menten derivations have been the most useful aspect of this chapter. 

As before, the mechanism gives rise to an overall third-order rate law, in agreement with experiment. Although this procedure is much simpler than the steady-state approach, it is less flexible it is more difficult to extend to more complex mechanisms and it is not so easy to establish the conditions under which the approximation is valid. 

Meiron (12) and Kessler et al. (13) have shown that numerical studies for small surface energy give indications of the loss-of-existence of the steady-state solutions. In these analyses numerical approximations to boundary integral forms of the freeboundary problem that are spliced to the parabolic shape far from the tip don t satisfy the symmetry condition at the cell tip when small values of the surface energy are introduced. The computed shapes near the tip show oscillations reminiscent of the eigensolution seen in the asymptotic analyses. Karma (14) has extended this analysis to a model for directional solidification in the absence of a temperature gradient. 

To verify that steady state catalytic activity had been achieved, the catalyst was allowed to operate uninterrupted for approximately 8 hours. The catalyst was then removed from the reactor and the surface investigated by XPS. The results are shown in Figure 2c. The two major changes in the XPS spectrun were a shift in the iron 2p line to 706.9 eV and a new carbon Is line centered at 283.3 eV. This combination of iron and carbon lines indicates the formation of an iron carbide phase within the XPS sampling volume.(J) In fact after extended operation, XRD of the iron sample indicated that the bulk had been converted to FecC2 commonly referred to as the Hagg carbide.(2) It appears that the bulk and surface are fully carbided under differential reaction conditions. 

---