Rate Expressions, Simplifying

If [D] is much greater than k i, the rate expression simplifies to... 

The first thing to note is that stoichiometric quantities of reactants were used in this investigation. Because the reaction rate expression simplifies when stoichiometric quantities of reactants are used, the equations developed earlier in this chapter cannot be applied directly in the solution of this problem. Thus we will have to derive appropriate relations in the course of our analysis. 

Furthermore, if destruction of the intermediate is rate limiting, i.e. k2 + k [T] < k, then the overall rate expression simplifies to Equation 9.11 ... 

From inspection of eqn. (46), one can see how, at high conversions or where cumyl hydroperoxide is added, the rate expression simplifies and kt - k3S. Similar results occur at high [RH] and low rates of initiation, since [R02- ] is proportional to Rl 2. 

We substitute these definitions into the rate expression, simplify, and integrate. d[A]... 

We shall now consider a simplifying approximation for the system A 2P. The reaction proceeds at a rate expressed in terms of S by Eq. (3-33). If the shift resulting from the concentration jump is small, the term AK l82 is negligible in comparison to (1 + 4 l [P](,)S. In that case, the solution is... 

This result is experimentally indistinguishable from the general form, Equation (10.12), derived in Example 10.1 using the equality of rates method. Thus, assuming a particular step to be rate-controlling may not lead to any simplification of the intrinsic rate expression. Furthermore, when a simplified form such as Equation (10.15) is experimentally determined, it does not necessarily justify the assumptions used to derive the simplified form. Other models may lead to the same form. 

The rate expression, e.g. Eq. (169), simplifies further since we can neglect the last term. If we also assume that the rate-limiting step is irreversible or that the product concentration is low, we only have to consider the forward reaction, and the rate reduces to... 

Using the concept of the rate-determining step significantly simplifies the overall rate expression. Therefore, it is widely used in the analysis of kinetie data, especially in the field of heterogeneous catalysis. 

The absolute rates of vaporization and condensation are evaluated from the rate expressions given in Section III,B. In the past, the rate of mass transfer (which is the net rate of phase change) has not been calculated from an understanding of the physics of the phase-change process at the interface. The rate is generally evaluated by applying some simplifying assumptions to the process, rather than from an expression in terms of the dependent variables of the model equations. 

These equations can be evaluated by using the rate expression for fnet given in Section III,B,3. As shown in that section, the results are equivalent to evaluating the absolute rate terms but the computation procedure is greatly simplified. 

Since both hydrogen in the solution and the product A are weakly adsorbed species, equilibrium constants ka and k, are very small, which leads to KACA 1 and kh CH 1. Thus, the rate expression for the debenzylation can be simplified as a conventional Langmuir-Hinshelwood model. 

CBo C 0andCco CA0, the only concentration that will change appreciably during the course of the experiment is that of species A. This situation implies that the above rate expression will simplify to the following form. 

In this section we discuss the mathematical forms of the integrated rate expression for a few simple combinations of the component rate expressions. The discussion is limited to reactions that occur isothermally in constant density systems, because this simplifies the mathematics and permits one to focus on the basic principles involved. We will again place a V to the right of certain equation numbers to emphasize that such equations are not general but are restricted to constant volume batch reactors. The use of the extent per unit volume in a constant volume system ( ) will also serve to emphasize this restriction. For constant volume systems,... 

The rate expression can be further simplified because in a real system the partial pressures of H20 and H2 are much higher than the partial pressures of CO and C02, which mean that the partial pressures of H20 and H2 are practically constant. 

The rate expression can be simplified by making use of the relationship between the fluorescence lifetime and the spectrum of the donor molecule and lumping together all the constants in one characteristic range R0. The rate of transfer is then... 

If DB quickly diffuses away ( 2 is large compared to k i), the expression simplifies to feobs = k and is solely dependent on the rate of deuteron abstraction. In other words, the isotopic exchange rate truly reflects the rate of deuteron abstraction. If k2 is relatively small, then the expression simplifies to feobs = and the... 

There is the further possibility that the transition C/ + C(0) C(CO)b is either slow (Case 1) or fast (Case 2) in comparison with C(CO)a — CO( ) -f- C/. The rate expression to be derived is the same in either case, but the interpretation of the individual rate constant, is, in Equation (5) will be different. When Case 1 holds, ja represents the rate constant for the surface rearrangement when Case 2 holds, ja represents the rate constant for the desorption of (C0) . It is not possible, on the basis of present experimental evidence, to decide which case is operative. It is conceivable that each case will be operative but in different temperature ranges. Assuming for the moment that Case 1 holds, the general expressions given above can be simplified to... 

In order to derive a simplified burning rate expression for the AP particles, it is assumed that for the area of fastest regression there is an effective region of thick-... 

Following Carbeny (1976), in this book the term rate coefficient is used for the proportionality coefficients kt in the typical rate expression of the form r] = kJ(C). To simplify the following analysis, a first-order elementary reaction is considered. Then the intrinsic reaction rate can be... 

The rate expression is a mathematical description of the rate of the reaction at any time t in terms of the concentrations(s) of the molecular species present at that time. By simplifying Equation (1) to... 

The second simplified form corresponds to the case where A, R and S are all weakly adsorbed, i.e. 1 KApA + KRpR + Ksps. Then eqn. (3) is reduced to a first-order rate expression... 

Unimolecular Reactions For the case of unimolecular reaction of A, that is, the reactant molecule in 10.84 is not present, the CTST expression for the rate constant simplifies to... 

It is common within the industry to characterize chemical processes in terms of one or a few global reaction steps, assigning an Arrhenius rate expression to describe the rate of each reaction. If knowledge of the detailed chemistry is inadequate or the chemical scheme is to be combined with computational fluid dynamics for a complex flow description, a simplified chemistry may be necessary. It is important, however, to realize that such a chemical description can only be used for the narrow range of conditions (temperature, composition, etc.) for which it is developed. Any extrapolation outside these conditions may be erroneous or even disastrous. 

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