The Steady-State Assumption

the factor 2 is included explicitly, this time because two radicals are consumed in each termination reaction. 

Reactions involving intermediates present in very very low, and almost constant concentrations, called steady state concentrations, are in the steady state region. As will be shown in Chapter 6, many chain and non-chain reactions involve such intermediates. 

If intermediates are present in low steady concentrations this implies that 

Note this is written as a rate of formation of the intermediate. This is the normal way of expressing the change in concentration of the intermediate with time. 

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The interpretations of Michaelis and Menten were refined and extended in 1925 by Briggs and Haldane, by assuming the concentration of the enzyme-substrate complex ES quickly reaches a constant value in such a dynamic system. That is, ES is formed as rapidly from E + S as it disappears by its two possible fates dissociation to regenerate E + S, and reaction to form E + P. This assumption is termed the steady-state assumption and is expressed as... 

Given the relationships between E, S, and I described previously and recalling the steady-state assumption that d[E /dt= 0, from Equations (14.14) and (14.16) we can write... 

Assuming that chain-breaking occurs exclusively by step 6, derive the steady-state rate equation for -r/[H2C204]/c/f. Make the steady-state assumption for Cl and CiO , and assume long chains (that is, [Fe2+] added [C121). The answer will involve R. Repeat the derivation for chain termination with step 7. 

Setting the derivatives with respect to time in Equations (Al) and (A2) equal to zero (i.e. invoking the steady state assumption), solving for [R°c] and [R1.J, and substituting into Equation (A3) gives ... 

Assuming that this process runs under steady state conditions, as for an industrial flow reactor with a constant inflow of reactants and a constant outflow of products, the concentration of the intermediate will be constant, as expressed in the steady-state assumption ... 

Application of the steady state assumption yields [O] overall rate becomes... 

Derive the rate expression for the formation of CH4 by using the steady state assumption. 

Under the steady-state assumption, the fraction of dose absorbed, Fa, is 1 - C0J... 

Because the steady-state assumption leads to the equilibrium relation for the bromine atom concentration (4.2.20), it does not matter what mechanism is assumed to be responsible for establishing this equilibrium. Alternative elementary reactions for the initiation and termination processes, which give rise to the same equilibrium relationship, would also be consistent with the observed rate expression for HBr formation. For example, the following reactions give rise to the same equilibrium ... 

As with most assumptions and approximations, those professors who do not ignore this entirely will undoubtedly think that you should at least know that it s an assumption. What the steady-state assumption actually... 

Schachter, H. (1972). The use of the steady-state assumption to derive kinetic formulations for the transport of a solute across a membrane. In Metabolic Transport, ed. Hokin, L. E., Metabolic Pathways. Vol. 6, Series ed. Greenberg, D. M., Academic Press, New York, pp. 1-15. 

We assumed that the adduct formed between the substrate and the active form of the catalyst obeys the steady-state assumption. Equations (6.225) to (6.227) are thus replaced by... 

Despite its predictive power and successful application on a variety of large-scale metabolic networks, stoichiometric analysis also encompasses a few inadequacies. In particular, stoichiometric analysis largely relies on the steady-state assumption and is not straightforwardly applicable to analyze complex time-dependent dynamics in metabolic systems. Similarly, stoichiometric analysis does not allow us to account for allosteric regulation, considerably delimiting its capabilities to predict dynamic properties. See also Section V.C for a discussion of the limits of stoichiometric analysis. 

The Michaelis-Menten equation can also be derived by applying the steady state assumption to the following scheme ... 

The steady-state assumption discussed in the consideration of the H2—Br2 chain system is applied for determination of the chain carrier concentration (R) ... 

The steady-state assumption is widely applied since the concentration of adduct-free myoglobin, PFe is very small. 

The method cannot be improved with respect to the steady state assumption, that is, the input air flows have to be constant during a test. However, the product flow of conversion gas can be transient and the method will still work if the measurement system s response rate is high enough. 

STEADY STATE TREATMENT. While the Michaelis-Menten model requires the rapid equilibrium formation of ES complex prior to catalysis, there are many enzymes which do not exhibit such rate behavior. Accordingly, Briggs and Haldane considered the case where the enzyme and substrate obey the steady state assumption, which states that during the course of a reaction there will be a period over which the concentrations of various enzyme species will appear to be time-invariant ie., d[EX]/dr s 0). Such an assumption then provides that... 

This fact reminds us that in making the steady state assumption, we are treating both E and EX as time-invariant forms.]... 

In using the equilibrium treatment, one should bear in mind that the rate laws so obtained are generally different in form from those derived by the steady-state assumption for the same mechanism. 

A mathematical simplification of rate behavior of a multistep chemical process assuming that over a period of time a system displays little or no change in the con-centration(s) of intermediate species (i.e., d[intermedi-ate]/df 0). In enzyme kinetics, the steady-state assumption allows one to write and solve the differential equations defining fhe rafes of inferconversion of various enzyme species. This is especially useful in initial rate studies. 

Equation 3-22 for the polymerization rate is not directly usable because it contains a term for the concentration of radicals. Radical concentrations are difficult to measure quantitatively, since they are very low ( 10-8 M), and it is therefore desirable to eliminate [M- from Eq. 3-22. In order to do this, the steady-state assumption is made that the concentration of radicals increases initially, but almost instantaneously reaches a constant, steady-state value. The rate of change of the concentration of radicals quickly becomes and remains zero during the course of the polymerization. This is equivalent to stating that the rates of initiation Rj and termination R, of radicals are equal or... 

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. 

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