Steady state approximation for

Using the Bodenstein steady state approximation for the intermediate enzyme substrate eomplexes derives reaetion rate expressions for enzymatie reaetions. A possible meehanism of a elosed sequenee reaetion is ... 

The concentration of the reaction intermediate AB may be determined by using the steady state approximation for intermediates,... 

Applying the rate expressions to Equations 1-222, 1-223, 1-224, 1-225 and 1-226, and using the steady state approximation for CH3, C2H5, and H, for a eonstant volume bateh reaetor yields ... 

Develop a suitable rate expression using the Michaelis-Menten rate equation and the quasi-steady-state approximations for the intermediate complexes formed. 

Making the steady-state approximation for [PFe], derive the rate law. Next, repeat the derivation including the reverse step with k-2. If [CO] and [02] are s> [PFe(O2)]0, what is the expression for ke, as defined in Chapter 3 ... 

Derive the rate law, making the steady-state approximation for the concentration of the intermediate (signified with an asterisk), which is a rearranged structure of the parent. For... 

Derive the expression shown for the rate constant for product buildup, making the steady-state approximation for the intermediates. It is best to use inventive shortcuts rather than tedious algebra. 

Steady-state mechanism. Derive the expression for -d[hR]/dt in this scheme, making the steady-state approximation for [A] and [B]. The answer must contain no concentration other than [AB],... 

One set of experiments was done with both Q and B present at initial concentrations much higher than that of A. With k, kx, and k-j known from other work, the value of k was then estimated, because under these conditions the steady-state approximation for [I] held. To check theory against experiment, one can also determine the products. In the case at hand, meaningful data could be obtained only when concentrations were used for which no valid approximation applies for the concentration of the intermediate. With kinsim, the final amount of each product was calculated for several concentrations. Figure 5-3 shows a plot of [P]o<4R] for different ratios of [B]o/[Q]o the product ratio changes 38-fold for a 51-fold variation in the initial concentration ratio. Had the same ratios of [B]o/tQ]o been taken, but with different absolute values, the indicated product ratios would not have stayed the same. Thus, this plot is for purposes of display only and should not be taken to imply a functional relationship between the quantities in the two axes. 

The reader can show that, with the steady-state approximation for [Tl2+], this scheme agrees with Eq. (6-14), with the constants k = k i and k = k j/k g. Of course, as is usual with steady-state kinetics, only the ratio of the rate constants for the intermediate can be determined. Subsequent to this work, however, Tl2+ has been generated by pulse radiolysis (Chapter 11), and direct determinations of k- and k g have been made.5... 

With the steady-state approximation for [V(OH)Cr4+], the rate law becomes... 

Derive the rate expression. Make the steady-state approximation for the radical intermediates, and assume that the chains are long. 

STRATEGY Construct the rate laws for the elementary reactions and combine them into the overall rate law for the decomposition of the reactant. If necessary, use the steady-state approximation for any intermediates and simplify it by using arguments based on rapid pre-equilibria and the existence of a rate-determining step. 

By applying the steady-state approximation for the concentration of the intermediate one obtains ... 

Note that it is assumed that reaction (1) is reversible, while reactions (2) and (3) are irreversible. Show, using steady-state approximations for the intermediates, that the overall reaction rate can be written as... 

Reaction (61) implies that the transition state contains 3 SCN ions. A steady state approximation for [ (SCN)2 ] leads to the observed rate law. A temperature variation study of /fj and is included in this paper. It was also concluded that simple redox breakdown of FeSCN is of negligible importance. 

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 quasistationary 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. 

Equation H also involves the concentration of a reaction intermediate [NO]. If we make the steady-state approximation for this species,... 

In 1919 Christiansen (25), Herzfeld (26), and Polanyi (27) all suggested the same mechanism for this reaction. The key factor leading to their success was recognition that hydrogen atoms and bromine atoms could alternately serve as chain carriers and thus propagate the reaction. By using a steady-state approximation for the concentrations of these species, these individuals were able to derive rate expressions that were consistent with that observed experimentally. 

However, the complete solution of these three simultaneous differential equations is difficult to obtain and is no more instructive than the approximate solution that can be obtained by means of the steady-state approximation for intermediates. If one sets the time derivatives in equations 4.2.17 and 4.2.18 equal to zero and adds these equations, their sum is found to be... 

Since collisional processes occur so rapidly, the concentration of the A molecules builds up to its steady-state value in a small fraction of a second and the steady-state approximation for A is appropriate for use. 

Reaction rate expressions for enzymatic reactions are usually derived by making the Bo-denstein steady-state approximation for the intermediate enzyme-substrate complexes. This is an appropriate assumption when the substrate concentration greatly exceeds that of the enzyme (the usual laboratory situation) or when there is both a continuous supply of reactant and a continuous removal of products (the usual cellular situation). 

In this equation, kjy is the rate constant for the diffusion-limited formation of the encounter complex, d is the rate constant for diffusion apart, and ka is that for the activation step, i.e. M-L bond formation. Based on the steady-state approximation for the encounter complex concentration, the apparent rate constant for the on reaction is kon = k kj (k - ,+ka), and the activation volume is defined as... 

Considering the rate equation v2 = k2[ES] and making use of the steady-state approximation for [ES], we obtain... 

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... 

With a steady state approximation for the chain-carrying intermediates in (2.59) and (2.60) and the assumption of long chain length... 

Consider a straight tubular runner of length L. A melt following the power-law model is injected at constant pressure into the runner. The melt front progresses along the runner until it reaches the gate located at its end. Calculate the melt front position, Z(f), and the instantaneous flow rate, Q t), as a function of time. Assume an incompressible fluid and an isothermal and fully developed flow, and make use of the pseudo-steady-state approximation. For a polymer melt with K = 2.18 x 10 N s"/m and n = 0.39, calculate Z(t) and Q(t)... 

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

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