Validity of Steady-State Assumption

Another consideration in the apphcation of the various kinetic expressions is the uncertainty in some reaction systems as to whether the initiator-coinitiator complex is soluble. Failure of the usual kinetic expressions to describe a cationic polymerization may indicate that the reaction system is actually heterogeneous. The method of handling the kinetics of heterogeneous polymerizations is described in Sec. 8-4c. 

The theoretical molecular weight distributions for cationic chain polymerizations are the same as those described in Sec. 3-11 for radical chain polymerizations terminating by reactions in which each propagating chain is converted to one dead polymer molecule, that is, not including the formation of a dead polymer molecule by bimolecular coupling of two propagating chains. Equations 2-86 through 2-89, 2-27, 2-96, and 2-97 with p defined by Eq. 3-185 

For polymerizations carried out to high conversions where the concentrations of propagating centers, monomer, and transfer agent as well as rate constants change, the polydispersity index increases considerably. Relatively broad molecular-weight distributions are generally encountered in cationic polymerizations. 

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Problem 6.12 Typical Tj values determined from photoinitiated radical chain polymerization with intermittent illumination are in the range O.l-lO s. Calculate from this the duration of the nonsteady-state period and comment on the validity of steady-state assumption made in radical chain polymerization. 

The steady-state approximation is a very useful assumption since, as shown below, it allows one to eliminate the inconvenient radical concentration term [M ] and find an expression for it in terms of known or measurable parameters. The validity of steady-state approximation has been shown experimentally in many polymerizations (see Problem below). 

In addition to the quasi-steady state assumption, the other assumptions required to arrive at equation (1) are 1. the aerosol itself does not coagulate 2. there is a fully developed concentration gradient around each aerosol particle and 3. the concentration of unattached radon progeny atoms is much greater than the concentration of aerosol particles (in order that concentration gradients of radon progeny atoms may exist). This last assumption is usually not valid since the radon progeny concentration is usually much less than the aerosol concentration. 

However, the validity of a similar assumption has to be questioned, in case the skin has previously been treated with a topically applied formulation [126], Opinions differ, whether the distinct curvature of the steady-state stratum corneum concentration gradient, reported in literature, may be an artifact of a wrong depth scale, since such a behavior cannot be reasonably explained by the established diffusion theory. 

Finally, one should recognize that determinations of the critical concentration depend wholly on the validity of the equilibrium or steady-state assumptions. If a stable end point for prdtomer-polymer coexistence is not attained, then kinetic factors affect the observed behavior. With the well recognized tendency of tubulin to lose its ability to engage in assembly reactions upon storage even at low temperature, and with the presence of various nucleotide hydrolases and transphosphorylases in microtubule protein, such kinetic effects are a serious problem. 

It is hardly surprising that blood phenytoin levels bear little relation to the total daily dose (B4, B5, Cl, G4, G6, L12), or that the concept of the steady-state blood levels frequently docs not hold in clinical practice (B7, M15). This indicates the necessity for regulating phenytoin therapy on the basis of blood concentrations, assuming that this bears a reasonably close correlation to the therapeutic effect. Conclusive evidence of the validity of this basic assumption is lacking (Table 1). 

The steady-state assumption is not unique to polymerization kinetics. It is often used in developing the kinetics of many small-molecule reactions that involve highly reactive intermediates present at very low concentrations—conditions that are present in radical chain polymerizations. The theoretical validity of the steady-state assumption has been discussed [Kondratiev, 1969] and its experimental validity shown in many polymerizations. Typical polymerizations achieve a steady-state after a period, which may be at most a minute. 

The measurement of xs also allows an evaluation of the validity of the usual steady-state assumption in radical chain polymerization from the relationship [Flory, 1953]... 

The time required for [M-] and Rp to reach their steady-state values is calculated as 65, 6.5, and 0.65 s, respectively, for r, values of 10, 1, and 0.1 s. Thus, in the usual polymerization the steady-state assumption is valid after a minute or so at most. 

Deviations can also occur when loss processes for 03 other than reaction (6) become significant so that 03 is no longer in a steady state. These additional loss processes may include photolysis of 03 as well as reactions with N02, alkenes, and the radicals H02 and OH. At sunset and sunrise, deviations are expected because the rate of photolysis of N02 is sufficiently small that steady-state assumptions are not valid (Calvert and Stockwell, 1983 Parrish et al., 1986). Finally, this ratio can be perturbed by fluxes of NO from the surface and micrometeorological effects (Carroll and Thompson, 1995). 

The membrane containing the immobilized enzyme is handled by partitioning it into a specified number of volume elements so that Equations 20.23 and 20.24 are valid in this model. While the concentration of each species may vary from element to element, the steady-state assumption (d[ES]/dt = 0) may be invoked independently for each volume element. This results in the definition of the Michaelis-Menton constant, KM ... 

The steady-state assumption is valid when the concentration ES of the enzyme-substrate complex ES is constant, and when the total enzyme concentration, tot, is small relative to that of the substrate, i.e. tot S. In most cases this assumption holds over a long period of the reaction, as illustrated in Fig. 5.10, which shows the significance of this assumption during reaction processes. 

Even this scheme represents a complex situation, for ES can be arrived at by alternative routes, making it impossible for an expression of the same form as the Michaelis-Menten equation to be derived using the general steady-state assumption. However, types of non-competitive inhibition consistent with the Michaelis-Menten type equation and a linear Linweaver-Burk plot can occur if the rapid-equilibrium assumption is valid (Appendix S.A3). In the simplest possible model, involving simple linear non-competitive inhibition, the substrate does not affect the inhibitor binding. Under these conditions, the reactions... 

The validity of the non-steady state assumption is shown in Figure 1, where the molecular weight of the copolymerization mixture increases with conversion at the beginning of copolymerization. The long lived radical is observed from the ESR spectrum in Figure 2, measured at room temperature for polymerization in toluene this agrees with the S02 radical similar to the norbomene-S02 system (16). 

Theoretical. The theory of steady state diffusion in a hollow sphere has been described by Crank (16). Because each frustum-shaped cell in the system closely approximates a spherical sector of a hollow sphere, a theoretical model can be developed on this basis to predict the release characteristics for this sytem. This assumption should be valid until the point is reached such that the curved interface (r in Figure 2) touches the flat impermeable backing, which should represent ca. 90% of the release. 

How does varying the substrate concentration relative to the enzyme concentration affect the validity of the steady-state assumption and the applicability of the Michaelis-Menten expression ... 

This is the same result as that obtained from the steady-state hypothesis (eq. 3.106). The validity of the steady-state assumption should be proved for each reaction to ensure consistency between the assumption and the experimental data. 

To determine whether phenylbromoacetylene follows step (b) exclusively or whether step (a) competes, the product ratios (PR) of diethylphenylethynylphos-phonate to phenylacetylene have been studied. For reaction (260), the PR is given by equation (261a). This relation is valid provided that the steady-state assumption... 

When both these assumptions are valid, at every point in time the temperature of the fluid is spatially uniform, and the wall temperature will be predicted by the equations valid for steady state. In a flat vessel wall the temperature changes linearly, however the straight line moves according to time. 

The first assumption is probably valid, since the other sources listed in Table 6.2 do not greatly alter the results derived by considering rivers alone. The issue of steady state cannot be verified for very long (millions of years) timescales, but the geological evidence does suggest that the concentration of major ions in seawater has remained broadly constant over very long time periods (Box 6.2). As an example of the residence time calculation, consider sodium (Na+) ... 

The build-up of measurable concentrations of an intermediate in a stepwise reaction does not occur in the majority of cases and the concentration can be very small compared with that of reactant and product. Under these conditions, the Bodenstein steady-state assumption is valid and Equation (2), the mechanism involving an intermediate, can be solved to give Equation (3). 

The theoretical validity of the steady-state assumption has been discussed [2] and its experimental validity has been shown in many polymerizations. 

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