Theory, monomolecular rate

Using the monomolecular rate theory developed by Wei and Prater, we have analyzed the kinetics of the liquid-phase isomerization of xylene over a zeolitic catalyst. The kinetic analysis is presented primarily in terms of the time-independent selectivity kinetics. With the establishment of the basic kinetics the role of intracrystalline diffusion is demonstrated by analyzing the kinetics for 2 to 4 zeolite catalyst and an essentially diffusion-free 0.2 to 0.4 m zeolite catalyst. Values for intracrystalline diffusivities are presented, and evidence is given that the isomerization is the simple series reaction o-xylene <= m-xylene <= p-xylene. 

The techniques of monomolecular rate theory easily transform measured reaction data into a form where we can analyze apparent kinetics and the effects of intracrystalline diffusion by the use of selectivity data. Time dependency has been eliminated. Since selectivity is extremely reproducible and is independent of short-term aging effects, the number of experimental runs is reduced while data reliability is maintained. For catalyst evaluation at any temperature, it is necessary to determine the equilibrium composition and the straight-line reaction path. With this information any catalyst can be evaluated at this temperature with simply the additional information from a curved-line reaction path. The approach used in the application of monomolecular rate theory to the xylene isomerization selectivity kinetics is as follows. Reference is made to the composition diagram, Figure 1. 

Applicability of Monomolecular Rate Theory to Xylene Isomerization Selectivity Kinetics over Fresh AP Catalyst. The kinetics of liquid-phase xylene isomerization over fresh zeolite containing AP catalyst are effectively interpreted by pseudomonomolecular rate theory. The agreement between the experimental data (data points) and predicted reaction paths (solid lines) for operation at 400° and 600°F is shown in Figure 2. The catalyst used was in the form of extrudates comprised of the zeolite component and an A1203 binder. Since xylene disproportionation to toluene and trimethylbenzenes was low, selectivity data were obtained by mere normalization of the xylene compositions (2 axyienes = 1.0). 

In this approach properties of potential energy surfaces are investigated from the point of view of all possible monomolecular transformations of the given reactants. A plausible suggestion concerning the mechanism of the reaction under study is usually made on the basis of reaction barriers or activation energies. Moreover, in some studies, partition functions are evaluated and rate constants are obtained within the framework of the absolute rate theory. 

Because a is a parameter that cannot be calculated from first principles. Equation 1-95 cannot be used to calculate reaction rate constant k from first principles. Furthermore, the collision theory applies best to bimolecular reactions. For monomolecular reactions, the collision theory does not apply. Tr3dng to calculate reaction rates from first principles for all kinds of reactions, chemists developed the transition state theory. 

In Fig. 6 the dependence of the reaction rate on the temperature is presented for a typical case T0 = 300°K, TB = 1900°K, activation heat A = 50,000 cal/mole. The three curves correspond to three different assumptions about the dependence of the reaction rate on the concentration independence (0), direct proportionality—a monomolecular reaction (I), and proportionality to the square of the concentration—a bimolecular reaction (II). As is clear from the drawing, in all the cases the reaction rate is large for temperatures close to the combustion temperature. This circumstance will form part of the foundation of a theory of flame velocity. 

Let us have a look at the thermodynamic approach [91], The reaction rate for a monomolecular dissociation mechanism depends in transition state theory on the partial pressure of the metal oxide PmcKgy If we incorporate the loss of atoms we... 

Pulse radiolysis of some scavenger solutions in water, intermediates spectra, and kinetics of their decay in liquid ammonia are investigated. Rate constant and activation energy are calculated for the latter. The dependence of the disappearance of intermediates on concentration is analyzed. It is shown that rate constant of reactions of pseudo-first order is not proportional to acceptors concentration. One of the possible reasons is that first order reaction was not taken into consideration. On this basis, rate constants of reactions with acceptors and these of monomolecular decay are calculated. It is revealed the decay of intermediates in 10 5-10 3M perchloric acid solutions does not depend upon HsO+ ion concentration. This fact is contrary to the present day theories about the nature of intermediates. 

In addition to illustrating many features of monomolecular reaction networks, Wei and Prater illustrated how these results, especially the straight line reaction paths, could be helpful in planning experiments for and the determination of rate constants, and this will be discussed later. Also, these same methods have been used in the stochastic theory of reaction rates, which consider the question of how simple macroscopic kinetic relations (e.g., the mass action law) can result from the millions of underlying molecular collisions—see Widom for comprehensive reviews [14]. 

Equations 56 and 57 apply to monomolecular reactions in the gas phase at the low pressure limit where the rate determining step is the activating collision. On the other hand, at the high pressure limit the theory of RRK predicts (see, e.g.. Ref. 986) = Eo (see footnote, p. 245). This explains the experimentally observed decrease of Ea with decreasing pressure. The decrease of Ea with temperature at the low pressure limit, predicted by Eq. 57, is less pronounced, and is more difficult to observe (see Addendum). 

In chemical kinetics, the reaction rates are proportional to concentrations or to some power of the concentrations. Phenomenological equations, however, require that the reaction velocities are proportional to the thermodynamic force or affinity. Affinity, in turn, is proportional to the logarithms of concentrations. Consider a monomolecular reaction between a reactant B and a product P in the vicinity of global equilibrium B = P. The rates of reaction by the kinetic theory are... 

The mathematical theory of the frequency-domain methods consists of beautiful applications of matrix and complex analysis. The general matrix rate equations have been derived for the monomolecular photochemical processes, and matrix analysis is used in deriving the general solution for the temporal concentrations of the excited species in the presence of an arbitrary functional form of excitation. The sinusoidal excitation and dual-phase lock-in detection of the emission lead to a signal which can be effectively treated as a complex number. For instance, the Kramers-Kronig relation, better known fi om the solid-state physics, can be used for checking the internal consistency of data. 

The first kinetic investigation [1], see Fluor Erg.-Bd. 1, 1959, p. 229, showed that OFg decomposes in a homogeneous monomolecular reaction between 523 and 543 K at some hundred Torr. The measured rate constant k was assumed to correspond to the activation process in reaction (1), see below (formation of OF2, followed by decomposition). The constant A=10 L mor s" found, is regarded to be a factor of six too high for a monomolecular reaction. On this basis it was supposed that the activation energy Ea = 39 was too high (this was not confirmed later, see below) or that chain reactions were involved [2]. The reaction is discussed in light of the theory of monomolecular reactions (see, e.g. [3, 4]). 

For a monomolecular reaction, in terms of the transition state theory (the RRKM approximation—Ref [14]), the total reaction rate obtained by... 

The rate constant becomes independent of pressure and can be computed using the transition-state theory expression. The reaction is first order in concentration [A], as expected for a monomolecular reaction. 

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