Steady-state, approximation rate constant

To be analytically useful equation 13.16 needs to be written in terms of the concentrations of enzyme and substrate. This is accomplished by applying the steady-state approximation, in which we assume that the concentration of ES is essentially constant. After an initial period in which the enzyme-substrate complex first forms, the rate of formation of ES... 

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 ... 

For Scheme XIV, and for each of the following sets of rate constants, calculate the exact relative concentration cb/ca as a function of time. Also, for each set, calculate the approximate values of cb/ca using both the equilibrium assumption and the steady-state approximation. 

Apply the steady-state approximation to Scheme XXII for ester hydrolysis to find how the experimental second-order rate constant qh is related to the elementary rate constants. 

If a reaction system consists of more than one elementary reversible reaction, there will be more than one relaxation time in general, the number of relaxation times is equal to the number of states of the system minus one. (However, even for multistep reactions, only a single relaxation time will be observed if all intermediates are present at vanishingly low concentrations, that is, if the steady-state approximation is valid.) The relaxation times are coupled, in that each relaxation time includes contributions from all of the system rate constants. A system of more than... 

Using the steady-state approximation, and taking into account the fact that the values of kQn are relatively insensitive to the ratio of radii rx/rB (Debye, 1942), so that one can set rx/rB l, one gets the expression in Scheme 3-33 as a good approximation for the encounter rate coefficient (R = gas constant). 

For simplicity, our initial discussion will deal only with situations where the steady-state approximation holds. If it happens that k2 k-, then the steady-state rate constant is... 

Table 4-1 lists six combinations of rate constants for which an RCS can be defined and two others lacking one. A method has been presented for exploring the concept of the RCS by means of reactant fluxes.11 Consider the case k < (k- + k2), such that the steady-state approximation is valid. One defines an excess rate , for each step i as the difference between the forward rate of that step and the net forward rate v/. Thus, for Step 1,... 

Reactant fluxes. Calculate values of , for the combination of rate constants in Tables 4-1 and 4-2 for those systems for which the steady-state approximation holds. Construct a diagram of the fluxes at the start of the reaction when [A]o = 1. 

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. 

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... 

Now we make the steady-state approximation, that any intermediate remains at a constant, low concentration. The justification for this approximation is that the intermediate is so reactive that it reacts as soon as it is formed. Because the concentration of the intermediate is constant, its net rate of formation is zero, and the previous equation becomes... 

In industry, as well as in a test reactor in the laboratory, we are most often interested in the situation where a constant flow of reactants enters the reactor, leading to a constant output of products. In this case all transient behavior due to start up phenomena have died out and coverages and rates have reached a constant value. Hence, we can apply the steady state approximation, and set all differentials in Eqs. (142)-(145) equal to zero ... 

The intermediate reaction complexes (after formation with rate constant, fc,), can undergo unimolecular dissociation ( , ) back to the original reactants, collisional stabilization (ks) via a third body, and intermolecular reaction (kT) to form stable products HC0j(H20)m with the concomitant displacement of water molecules. The experimentally measured rate constant, kexp, can be related to the rate constants of the elementary steps by the following equation, through the use of a steady-state approximation on 0H (H20)nC02 ... 

The H20 exchange mechanism was studied by Hunt et al. (32) who reported that exchange between aqueous solvent and Fein(TPPS)(H20)2 occurs with a first-order rate constant (kex = 1.4xl07s-1 in water at 298 K) far exceeding the k0 s values determined at any [NO]. If the steady state approximation was applied with regard to the intermediate Fem(Por)(H20), the kohs for the exponential relaxation of the non-equilibrium mixture generated by the flash photolysis experiment would be,... 

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... 

Since the enzymatic reactions are generally rapid, it may be assumed that the steady-state approximation applies. Note, however, that although true is most systems, this is not always the case, as exemplified in Section 5.2.5. Each half-reaction is characterized by three rate constants, defined in Scheme 5.1. They may alternatively be characterized by the following... 

Similar to irreversible reactions, biochemical interconversions with only one substrate and product are mathematically simple to evaluate however, the majority of enzymes correspond to bi- or multisubstrate reactions. In this case, the overall rate equations can be derived using similar techniques as described above. However, there is a large variety of ways to bind and dissociate multiple substrates and products from an enzyme, resulting in a combinatorial number of possible rate equations, additionally complicated by a rather diverse notation employed within the literature. We also note that the derivation of explicit overall rate equation for multisubstrate reactions by means of the steady-state approximation is a tedious procedure, involving lengthy (and sometimes unintelligible) expressions in terms of elementary rate constants. See Ref. [139] for a more detailed discussion. Nonetheless, as the functional form of typical rate equations will be of importance for the parameterization of metabolic networks in Section VIII, we briefly touch upon the most common mechanisms. 

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 general, the effective overall rate constant associated with loss of reactants can be expressed in terms of the individual rate constants a-e in eq 3 by use of the steady state approximation. Simpler expressions can be obtained if the species related by diffusion (a,b) and activation (c,d) processes are assumed to be in thermal equilibrium. In such a case one finds straightforwardly that the effective first order rate constant, k (r), for electron transfer at separation r can be written as... 

It can be solved by the so-called Bodenstein or steady-state approximation. This approximation supposes that the concentration of the reactive intermediate, in this case MS, is always small and constant. For a catalyst, of which the concentration is always small compared to the substrate concentration, it means that the concentration of MS is small compared to the total M concentration. The rate of production of products for the scheme in Figure 3.1 is given by equation (3). Equation (4) expresses the steady state approximation the amounts of MS being formed and reacting are the same. Equation (5) gives [M] and [MS] in measurable quantities, namely the total amount of M (Mt) that we have added. If we don t add this term, the nominator of equation (6) will not contain the term of k and the approximations that follow cannot be carried out. 

In Equation 11.9 we reserve the missing rate constant k4 for an elaboration of the mechanism). Following Briggs and Haldane we make the assumption that the steady-state approximation applies to ES and EP complexes ... 

Consequently, it is seen, from the measurement of the overall reaction rate and the steady-state approximation, that values of the rate constants of the intermediate radical reactions can be determined without any measurement of radical concentrations. Values k exp and xp evolve from the experimental measurements and the form of Eq. (2.31). Since (ki/k5) is the inverse of the equilibrium constant for Br2 dissociation and this value is known from thermodynamics, k2 can be found from xp. The value of k4 is found from k2 and the equilibrium constant that represents reactions (2.2) and (2.4), as written in the H2 Br2 reaction scheme. From the experimental value of k CX(l and the calculated value of k4, the value k3 can be determined. 

As will be discussed in the following chapter, most combustion systems entail oxidation mechanisms with numerous individual reaction steps. Under certain circumstances a group of reactions will proceed rapidly and reach a quasi-equilibrium state. Concurrently, one or more reactions may proceed slowly. If the rate or rate constant of this slow reaction is to be determined and if the reaction contains a species difficult to measure, it is possible through a partial equilibrium assumption to express the unknown concentrations in terms of other measurable quantities. Thus, the partial equilibrium assumption is very much like the steady-state approximation discussed earlier. The difference is that in the steady-state approximation one is concerned with a particular species and in the partial equilibrium assumption one is concerned with particular reactions. Essentially then, partial equilibrium comes about when forward and backward rates are very large and the contribution that a particular species makes to a given slow reaction of concern can be compensated for by very small differences in the forward and backward rates of those reactions in partial equilibrium. 

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. 

Thus we see that the pseudo-steady-state approximation gives orders of the reaction as the thermodynamic equilibrium approximation, the only difference being the definition of the rate constant... 

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. 

Multiscale ensembles of reaction networks with well-separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors ( modes ) is presented. In particular, we prove that for systems with well-separated constants eigenvalues are real (damped oscillations are improbable). For systems with modular structure, we propose the selection of such modules that it is possible to solve the kinetic equation for every module in the explicit form. All such solvable networks are described. The obtained multiscale approximations, that we call dominant systems are... 

The steady-state approximation on the base of the rule (13) is a linear function of the restored-and-cut cycles rate-limiting constants. It is the first-order approximation. 

Each reaction step has a forward rate constant. This model maps onto the steady state approximation. Section ref sec solve/steady. ... 

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