Rate equation coupled

Knowledge of the fornr of the rate equation, coupled with the experimental determination of the value of the rate constant, k, and the order, n, are valuable in a number of ways. Industrial chemists use this information to establish optimum conditions for preparing a product in the shortest practical time. The design of an entire manufacturing facility may, in part, depend on the rates of the critical reactions. 

Onr discussion of the free energy driving forces will mostly follow the discussions ontlined in Venables et al. [5] and in Markov [3], The extension of the atomistic approach to the organics will come from Verlaak et al. [15]. The elements of the rate equation will follow Venables et al. [5]. Dynamic scaling will then follow as an extension to the rate equations coupled with some assumptions about the island size distributions [23]. For deposition beyond the first layer the comparison of model predictions with experiment will follow Cohen et al. [29]. The height-height correlation description will follow Krim and Palasantzas [30]. 

The fonn of the classical (equation C3.2.11) or semiclassical (equation C3.2.11) rate equations are energy gap laws . That is, the equations reflect a free energy dependent rate. In contrast with many physical organic reactivity indices, these rates are predicted to increase as -AG grows, and then to drop when -AG exceeds a critical value. In the classical limit, log(/cg.j.) has a parabolic dependence on -AG. Wlren high-frequency chemical bond vibrations couple to the ET process, the dependence on -AG becomes asymmetrical, as mentioned above. 

Sets of first-order rate equations are solvable by Laplace transform (Rodiguin and Rodiguina, Consecutive Chemical Reactions, Van Nostrand, 1964). The methods of linear algebra are applied to large sets of coupled first-order reactions by Wei and Prater Adv. Catal., 1.3, 203 [1962]). Reactions of petroleum fractions are examples of this type. 

The appearance of Cd in the denominator means that D is coupled to a reversible step prior to the rds. If k, and k- were so large that the fast preequilibrium assumption is valid, then the Cd term in the denominator would drop out, and we would have v = A 2 CaObCd, giving the composition of the rds transition state. If k2 is very much larger than it, and k i, Eq. (5-59) becomes v = A CaOb the first step is now the rds, and the rate equation gives the transition state composition. 

In contrast to consecutive reactions, with parallel competitive reactions it is possible to measure not only the initial rate of isolated reactions, but also the initial rate of reactions in a coupled system. This makes it possible to obtain not only the form of the rate equations and the values of the adsorption coefficients, but also the values of the rate constants in two independent ways. For this reason, the study of mutual influencing of the reactions of this type is centered on the analysis of initial rate data of the single and coupled reactions, rather than on the confrontation of data on single reactions with intergal curves, as is usual with consecutive reactions. 

The results obtained showed, again, that the form of the rate equations and the values of their constants, obtained by the study of isolated reactions, are valid also in the coupled system. This was also confirmed by the observed agreement between the calculated and the experimental integral data (94)- Kinetic results and the analysis of the effect of reaction products revealed that adsorption of the reaction components was competitive and that all the compounds involved in the three reactions were adsorbed on the same sites of the catalytic surface. 

The usual chemical kinetics approach to solving this problem is to set up the time-dependent changes in the reacting species in terms of a set of coupled differential rate equations [5,6]. 

In general, solutions are obtained by coupling the basic conservation equation for the batch system, Eq. (16-49) with the appropriate rate equation. Rate equations are summarized in Tables 16-11 and 16-12 for different controlling mechanisms. 

Now, let us discuss the rate equations embodied in eq.(74). To do this, there is need of a statistical analysis. If the system is kept coupled to a thermostat at absolute temperature T, and assuming that w(i - >if) contains effects to all orders in perturbation theory, the rate of this unimolecular process per unit (state) reactant concentration k + is obtained after summation over the if-index is carried out with Boltzman weight factors p(if,T) ... 

There are a couple of ways that you might interpret the data above in order to determine the rate equation. If the numbers involved are simple, then one can reason out the orders of reaction. You can see that in going from experiment 1 to experiment 2, the [NO] doubles ([02] held constant) and the rate increased fourfold. This means that the reaction is second order with respect to NO. Comparing experiments 1 and 3, you see that the [02] doubles ([NO] was held... 

Wegner also treated the case wherein assembly is coupled to nucleotide hydrolysis. Here, we consider a slight modification of his model to deal with the microtubule process. Normally, the concentration of GTP is maintained by use of a GTP-regenerating system (MacNeal et ai, 1977), and the system at the steady-state plateau of assembly can be described as in Scheme II. Under these conditions, the assembly-disassembly reactions are no longer reversible, and the primed rate constants are used to emphasize that we are dealing with a different case. The rate equations for the two ends are now given as ... 

In this chapter we deal with single reactions. These are reactions whose progress can be described and followed adequately by using one and only one rate expression coupled with the necessary stoichiometric and equilibrium expressions. For such reactions product distribution is fixed hence, the important factor in comparing designs is the reactor size. We consider in turn the size comparison of various single and multiple ideal reactor systems. Then we introduce the recycle reactor and develop its performance equations. Finally, we treat a rather unique type of reaction, the autocatalytic reaction, and show how to apply our findings to it. 

Transition state theory yields rate coefficients at the high-pressure limit (i.e., statistical equilibrium). For reactions that are pressure-dependent, more sophisticated methods such as RRKM rate calculations coupled with master equation calculations (to estimate collisional energy transfer) allow for estimation of low-pressure rates. Rate coefficients obtained over a range of temperatures can be used to obtain two- and three-parameter Arrhenius expressions ... 

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