Steady-state approximation applicability

Fig. 8-10. Time constants Tor the approach to the generalized Henry s law gas-solution equilibrium for cloud drops and raindrops with radius r, (T = 293 1C assumed). The numbering refers to the following cases. (1) Mixing by diffusion inside the drop the same curve is obtained for diffusive transport in the gas phase for a highly soluble gas and a gas-liquid volume ratio of 106 (2) Case (1) calculated with Djf replacing Dg. (3) Steady-state approximation applicable to slightly soluble gases, H = 10. (4), (5) The same for H = 100 and H = 103, respectively. (6) Free-fall time for a drop over a distance of 100 m.

The overall rate of a chain process is determined by the rates of initiation, propagation, and termination reactions. Analysis of the kinetics of chain reactions normally depends on application of the steady-state approximation (see Section 4.2) to the radical intermediates. Such intermediates are highly reactive, and their concentrations are low and nearly constant throughout the course of the reaction ... 

The result of the steady-state condition is that the overall rate of initiation must equal the total rate of termination. The application of the steady-state approximation and the resulting equality of the initiation and termination rates permits formulation of a rate law for the reaction mechanism above. The overall stoichiometry of a free-radical chain reaction is independent of the initiating and termination steps because the reactants are consumed and products formed almost entirely in the propagation steps. 

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. 

Consider further Scheme XIV and rate equations (3-139) to (3-141). Evidently Cb will be small relative to (Ca + Cc) if ( -i + 2) i- Then B plays the role of a reactive intermediate in the overall reaction A— C. This is the usual condition that is taken as a warrant for the application of the steady-state approximation. If Cb is small, it is reasonable that Cb will be small throughout most the reaction, so it is set equal to zero. As Wong (53) has pointed out, however, the condition Cb = 0 is a sufficient but unnecessary condition for Eq. (3-142) to hold. Erom Eq. (3-140) we obtain... 

Several features of this treatment are of interest. Compare the denominators of Eqs. (3-147) and (3-149) Miller has pointed out that the form of Eq. (3-147) is usually seen in chemical applications of the steady-state approximation, whereas the form of Eq. (3-149) appears in biochemical applications. The difference arises from the manner in which one uses the mass balance expressions, and this depends upon the type of system being studied and the information available. 

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

Suppose in Scheme XVI that the steady-state approximation is applicable to the intermediate A. ... 

Application of a steady state approximation (that / t = / j, eq. 2) and a long chain approximation (negligible monomer consumption in the initiation or reinitiation steps) provides a number of useful relationships. ... 

Given the postulated reaction scheme, the net rate of reaction often takes a simple form when it is expressed in terms of the concentration of the intermediate. Such an expression is algebraically correct, and is the form one needs so as to propose and interpret the mechanism. This form is, however, usually not useful for the analysis of the concentration-time curves. In such an expression the reaction rate is given in terms of the concentration of the intermediate, which is generally unknown at the outset. To eliminate the concentration term for the intermediate, one may enlist certain approximations, such as the steady-state approximation. This particular method is applicable when the intermediate remains at trace levels. 

Application of the steady state approximation to the energized intermediate A gives the concentration of this elusive species ... 

The authors propose that AgO oxidises water to H2O2 which is rapidly oxidised in turn by Ag(II) to H02 , itself oxidised by further Ag(II) to O2. Application of the steady-state approximation to [Ag(nr)], [AgO ], [H2O2] and [HO2 ] produces the observed rate law (A). 

Application of the steady-state approximation leads to the observed kinetics. 

There are several assumptions inherent in the HMD multicomponent derivation that should be understood. In the model derivation, the authors made a steady-state approximation, which means that the solution is applicable for Cases I and II only after the surface films have formed. The model is not applicable during the initial early dissolution phase. HMD also point out that the time re-... 

The exchange proreeds in three steps. The first step is substitution of the free ligand at one end of the chelate to give the intermediate I(fca). The second is intramolecular proton transfer between the unidentate ligand in l(kb). The third is the reverse of the first (k a). Consequently, application of the steady-state approximation to the intermediate 1, whose concentration is reasonably assumed to be very low, provides... 

FIGURE 5.24. Variation of the voltammetric peak or plateau current with the concentration of H202 obtained in the same conditions as in Figure 5.23. Solid circles experimental values. Steady-state equations application of equation (5.28) (dotted line), equation (5.27) (dashed-dotted line) of equation (5.28) (dashed line). Simulation after removal of the steady-state approximation (solid line) with k3r° — 0.029 cm s 1, K3M = 37 pM, k4/ks = 0.0144, k6/k5 = 4.8 pM, kx = 1.7 x K ArV1, KiM = 128 pM, Ds — 1.5 x 5 cm2s 1. k4 — 30 M 1 s 1. Adapted from Figure 4 of reference 23, with permission from the American Chemical Society. 

Assuming for simplicity that only a particular base B is an effective catalyst in equation 1, application of the steady-state approximation derives in equation 2 the expression of the second-order rate constant, k, at a given concentration of B. 

In dealing with a complex reaction scheme as the one indicated above, one frequently introduces a so-called steady state approximation for reactive intermediates in order to find simplified rate laws (see, for example, Section 11.2). This approximation is usually sufficiently valid to give rise to useful results most physical chemistry texts discuss and use this application. In the steady state approximation for A, one writes... 

In order to better understand the detailed dynamics of this system, an investigation of the unimolecular dissociation of the proton-bound methoxide dimer was undertaken. The data are readily obtained from high-pressure mass spectrometric determinations of the temperature dependence of the association equilibrium constant, coupled with measurements of the temperature dependence of the bimolecular rate constant for formation of the association adduct. These latter measurements have been shown previously to be an excellent method for elucidating the details of potential energy surfaces that have intermediate barriers near the energy of separated reactants. The interpretation of the bimolecular rate data in terms of reaction scheme (3) is most revealing. Application of the steady-state approximation to the chemically activated intermediate, [(CH30)2lT"], shows that. 

Application of the steady-state approximation to the initially formed, chemically activated complex yields the rate equation for disappearance of the bare chloride ion and formation of the collisionally stabilized Sfj2 intermediate. Equation (7). The apparent bimolecular rate constant for the formation of the stabilized complex... 

As carried out above for the Lindemann mechanism, application of the steady-state approximation gives the apparent unimolecular rate constant in Equation (24) where [Av] represents the IR photon density. Again two limits may be considered. 

Equation (48) e ees with experimental results in some circumstances. This does not mean the mechanism is necessarily correct. Other mechanisms may be compatible with the experimental data and this mechanism may not be compatible with experiment if the physical conditions (temperature and pressure etc.) are changed. Edelson and Allara [15] discuss this point with reference to the application of the steady state approximation to propane pyrolysis. It must be remembered that a laboratory study is often confined to a narrow range of conditions, whereas an industrial reactor often has to accommodate large changes in concentrations, temperature and pressure. Thus, a successful kinetic model must allow for these conditions even if the chemistry it portrays is not strictly correct. One major problem with any kinetic model, whatever its degree of reality, is the evaluation of the rate cofficients (or model parameters). This requires careful numerical analysis of experimental data it is particularly important to identify those parameters to which the model predictions are most sensitive. 

The first example we wiU use as an application of the equilibrium step approximation and pseudo-steady-state approximation is the reaction... 

An indirect method has been used to determine relative rate constants for the excitation step in peroxyoxalate CL from the imidazole (IM-H)-catalyzed reaction of bis(2,4,6-trichlorophenyl) oxalate (TCPO) with hydrogen peroxide in the presence of various ACTs . In this case, the HEI is formed in slow reaction steps and its interaction with the ACT is not observed kinetically. However, application of the steady-state approximation to the reduced kinetic scheme for this transformation (Scheme 6) leads to a linear relationship of l/direct measure of the rate constant of the excitation step. 

Application of the usual steady-state approximation to this mechanism reveals that... 

This ratio provides the criterion for the applicability of the quasi-steady-state approximation for the concentration as shown in Table V. 

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

At a temperature of 407 °C (or, very nearly, at 422.5 °C) the assumption that A AT may be used as a fair approximation for AAG and application of the standard steady-state approximation for the trimethylene intermediate leads to an overall estimation of one-... 

Application of the steady-state approximation for Cl atoms results in the expressions A[C1] kesc Ci = k [C1][RH], which may be rearranged to the expression... 

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. 

Here also, then, we see that application of the steady-state approximation to the concentration of B when (k2 and/or k-1) y> k leads to a simple first-order concentration-time relationship for the formation of the product C, i.e. very much simpler than that in Equations 4.8. 

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