Rate-limiting step equations

The addition of the lanthanide hydride to the cyclohexene double bond is the rate-limiting step (Equation (19)),... 

Finally, we present the results of the case studies for Eley-Rideal and LH reaction mechanisms illustrating the practical aspects (i.e. convergence, relation to classic approximations) of application of this new form of reaction rate equation. One of surprising observations here is the fact that hypergeometric series provides the good fit to the exact solution not only in the vicinity of thermodynamic equilibrium but also far from equilibrium. Unlike classical approximations, the approximation with truncated series has non-local features. For instance, our examples show that approximation with the truncated hypergeometric series may supersede the conventional rate-limiting step equations. For thermodynamic branch, we may think of the domain of applicability of reaction rate series as the domain, in which the reaction rate is relatively small. 

Similar mechanisms are assigned to the reaction of C102 with Br02 and the disproportionation of C102. The reverse of the rate-limiting step (Equation 9.28) is formally analogous to the reductive cleavage mechanism discussed below in Section 9.15. 

Regeneration. The oxidized nicotinamide cofactors (NAD(P) ) are considerably more difficult to work with than ATP, but are more tractable than the reduced nicotinamide cofactors (NAD(P)H). The oxidized cofactors are sensitive to nucleophiles (8), but are relatively stable at pH 7 the reduced cofactors decompose by acid-catalyzed processes involving protonation at C-5 of the dihydropyridine ring as the rate-limiting step (equation iii) (9,10). 

The overall requirement is 1.0—2.0 s for low energy waste compared to typical design standards of 2.0 s for RCRA ha2ardous waste units. The most important, ie, rate limiting steps are droplet evaporation and chemical reaction. The calculated time requirements for these steps are only approximations and subject to error. For example, formation of a skin on the evaporating droplet may inhibit evaporation compared to the theory, whereas secondary atomization may accelerate it. Errors in estimates of the activation energy can significantly alter the chemical reaction rate constant, and the pre-exponential factor from equation 36 is only approximate. Also, interactions with free-radical species may accelerate the rate of chemical reaction over that estimated solely as a result of thermal excitation therefore, measurements of the time requirements are desirable. 

Using the conservation and balance equations for the active centers, but without the assumption of a rate-limiting step, the mathematically rigorous rate expression is the UCKRON-1 Test Problem given below. 

The kinetics of the contributory rate processes could be described [995] by the contracting volume equation [eqn. (7), n = 3], sometimes preceded by an approximately linear region and values of E for isothermal reactions in air were 175, 133 and 143 kJ mole-1. It was concluded [995] that the rate-limiting step for decomposition in inert atmospheres is NH3 evolution while in oxidizing atmospheres it is the release of H20. A detailed discussion of the reaction mechanisms has been given [995]. Thermal analyses for the decomposition in air [991,996] revealed only the hexavanadate intermediate and values of E for the two steps detected were 180 and 163 kJ mole-1. 

The reaction of Si02 with SiC [1229] approximately obeyed the zero-order rate equation with E = 548—405 kJ mole 1 between 1543 and 1703 K. The proposed mechanism involved volatilized SiO and CO and the rate-limiting step was identified as product desorption from the SiC surface. The interaction of U02 + SiC above 1650 K [1230] obeyed the contracting area rate equation [eqn. (7), n = 2] with E = 525 and 350 kJ mole 1 for the evolution of CO and SiO, respectively. Kinetic control is identified as gas phase diffusion from the reaction site but E values were largely determined by equilibrium thermodynamics rather than by diffusion coefficients. 

Equations 4.31 and 4.32 also suggest another important fact regarding NEMCA on noble metal surfaces The rate limiting step for the backspillover of ions from the solid electrolyte over the entire gas exposed catalyst surface is not their surface diffusion, in which case the surfacediffusivity Ds would appear in Eq. 4.32, but rather their creation at the three-phase-boundaries (tpb). Since the surface diffusion length, L, in typical NEMCA catalyst-electrode film is of the order of 2 pm and the observed NEMCA time constants x are typically of the order of 1000 s, this suggests surface diffusivity values, Ds, of at least L2/t, i.e. of at least 4 10 11 cm2/s. Such values are reasonable, in view of the surface science literature for O on Pt(l 11).1314 For example this is exactly the value computed for the surface diffusivity of O on Pt(lll) and Pt(100) at 400°C from the experimental results of Lewis and Gomer14 which they described by the equation ... 

On the fuel-lean side (kadPo2 kpc2H4) the oxygen coverage is near unity and step (8.3) is the rate limiting step (rls). Thus equation (8.1) reduces to ... 

We will list the elementary steps and decide which is rate-limiting and which are in quasi-equilibrium. For ammonia synthesis a consensus exists that the dissociation of N2 is the rate-limiting step, and we shall make this assumption here. With quasi-equilibrium steps the differential equation, together with equilibrium condition, leads to an expression for the coverage of species involved in terms of the partial pressures of reactants, equilibrium constants and the coverage of other intermediates. 

The hydrogenation of dioxomethylene, step (33) is, most likely, the rate-limiting step, although the hydrogenation of formate in (32) is a also candidate. By assuming that Eqs. (24), (23) and (29) are slow for the water-gas shift reaction and that (33) is slow for methanol synthesis, we arrive at the following set of equations, in which one site is assumed to consist of two copper atoms ... 

Propose a mechanism where the rate-limiting step is the recombination of adsorbed carbon C and adsorbed oxygen O and write up an equation for the rate. In the following we assume that only one adsorbate dominates the surface. The so-called MARI for the most abundant reaction intermediate. Here we assume that it is oxygen O. Is that reasonable ... 

When diffusion becomes the rate-limiting step, the time required for falling-rate drying can be estimated using an approximate equation ... 

The rate of anhydride addition, see Equation 10, is also the rate limiting step. 

By contrast, hydrolysis of the tertiary halide 2-chloro-2-methyl-propane (3,t-butyl chloride) in base is found kinetically to follow equation [2], i.e. as the rate is independent of [eOH], this can play no part in the rate-limiting step. This has been interpreted as indicating that the halide undergoes slow ionisation (in fact, completion of the R->-Cl polarisation that has already been shown to be present in such a molecule) as the rate-limiting step to yield the ion pair R Cle (4) followed by rapid, non rate-limiting attack by eOH or, if that is suitable, by solvent, the latter often predominating because of its very high concentration ... 

A kinetic distinction between the operation of the SN1 and SN2 modes can often be made by observing the effect on the overall reaction rate of adding a competing nucleophile, e.g. azide anion, N3e. The total nucleophile concentration is thus increased, and for the SN2 mode where [Nu ] appears in the rate equation, this will result in an increased reaction rate due to the increased [Nut]. By contrast, for the Stfl mode [Nut] does not appear in the rate equation, i.e. is not involved in the rate-limiting step, and addition of N3e will thus be without significant effect on the observed reaction rate, though it will naturally influence the composition of the product. 

If some of the processes prior to the rate limiting step are characterized by times comparable to that of the rate determining step, equation 4.1.4 may be rewritten as... 

However, if reaction 3 is rate limiting we can deduce something useful and we will illustrate the quasi-equilibrium method by using it to derive the kinetic equation under these conditions. This method assumes that all reactions prior to the rate limiting step are in equilibrium. Thus, for reaction 1 ... 

If q is negligibly small a stagnant diffusion layer in aqueous phase is a rate-limiting step. This case is often the most useful from an analytical point of view. The well-known equation for a reversible polarographic wave can be obtained as ... 

Despite the authority apparent in its name, no single rate law describes how quickly a mineral precipitates or dissolves. The mass action equation, which describes the equilibrium point of a mineral s dissolution reaction, is independent of reaction mechanism. A rate law, on the other hand, reflects our idea of how a reaction proceeds on a molecular scale. Rate laws, in fact, quantify the slowest or rate-limiting step in a hypothesized reaction mechanism. 

Taking the rate limiting step in the electron transport chain to be trans-membrane proton translocation, which occurs about five times per sulfate consumed (Rabus et al., 2006), the average stoichiometric number x (entered into REACT as to = 1/x) for Reaction 18.7 is five. Sulfate reducers conserve about 45 kJ mol-1 of sulfate consumed (Qusheng Jin, unpublished data), so we set AGp to this value and m to one. From equations 18.12 and 18.14, then, we can write... 

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