Frequency factor, definition

Usually one deals with a system whose equations of motion are invariant under time reversal, and the definitions of the dividing surface and reactant and product regions involve only coordinates, not momenta. Under these conditions (which will henceforth be assumed) the factor ux-(ux>0) in eqs. 4 and 5 can be replaced by lu J, and the frequency factor (and conversion coefficient) will be the same in the forward and backward directions, because every successful forward trajectory is the reverse of an equiprobable successful backward trajectory. One can then use a third form of the function, viz. 

Assume that equilibrium is maintained between and the reactants despite a unidirectional decomposition of M+. Then if the activated complex, M+, is regarded as an ordinary molecule, possessing the usual thermodynamic properties, with the exception that motion in one direction, i.e. along the reaction coordinate, leads to decomposition at a definite rate (Glasstone et al., 1941b), it can be shown hy classical statistical methods that the rate of decomposition of is equal to kTjh, a universal frequency factor dependent only on temperature and... 

Table 10.2 presents the kinetic information for the main reactions, in which the frequency factors have been calculated from turnover-frequency (TOF) data [8, 9]. This term, borrowed from enzymatic catalysis, quantifies the specific activity of a catalytic center. By definition, TOF gives the number of molecular reactions or catalytic cycles occurring at a center per unit of time. For a heterogeneous catalyst the number of active centers can be found by means of sorption methods. Let us consider that the active sites are due to a metal atom. By definition [15] we have ... 

In the first and second equation, E is the energy of activation. In the first equation A is the so-called frequency factor. In the second equation AS is the entropy of activation, the interatomic distance between diffusion sites, k Boltzmann s constant, and h Planck s constant. In the second equation the frequency factor A is expressed by means of the universal constants X2 and the temperature independent factor eAS /R. For our purposes AS determines which fraction of ions or atoms with a definite energy pass over the energy barrier for reaction. 

It is quite simple to say that this article deals with Chemical Dynamics. Unfortunately, the simplicity ends here. Indeed, although everybody feels that Chemical Dynamics lies somewhere between Chemical Kinetics and Molecular Dynamics, defining the boundaries between these different fields is generally based more on sur-misal than on knowledge. The main difference between Chemical Kinetics and Chemical Dynamics is that the former is more empirical and the latter essentially mechanical. For this reason, in the present article we do not deal with the details of kinetic theories. These are reviewed excellently elsewhere " The only basic idea which we retain is the reaction rate. Thus the purpose of Chemical Dynamics is to go beyond the definition of the reaction rate of Arrhenius (activation energy and frequency factor) for interpreting it in purely mechanical terms. 

The kinetic parameters for the n order kinetic model have been obtained using these definitions of reactivity for the pure steam gasification experiments of birch. All the activation energies lie between 228-238 kJ/mol and the reaction orders between 0.54 and 0.58, apart from definition 3. The frequency factors are somewhat more scattered, lying between 5-10 and 3-10 . Regarding the uncertainty of the calculation, definitions 2, 5 and 4 seem to give more precise results and it is interesting to notice that the error of the reaction order calculation does not depend on how a representative reactivity value is defined. 

It is very important to analyse the influence of the reactivity definition (eqn. 8 and 9) on the kinetic parameters. Since all representative reactivity definitions are related to a fixed degree of conversion (or a fixed interval), the difference between r and r will be a multiplying factor, independent of temperature and pressure, and therefore absorbed in the frequency factor. This means that whether equation 8 or 9 is used, the activation energy and the reaction order calculation will give the same result. 

Most of these questions are not important beyond the problem of comparing results from different laboratories, since many of the space time definitions vary from each other by just a calculable constant. However, in sane cases other issues may be involved, such as the assumed volume of vaporized liquid feed at inlet or at STP conditions, the definition of catalyst density, and so on, making comparison between results from various laboratories difficult and raising the potential for distorting the calculated activation energies and frequency factors of a reaction. There is no generally accepted convention for the definition of space time and the best we can expect is to know exactly how it has been defined in each case. There is an urgent need for this information and it must be demanded of authors by referees and editors if we are to remove some of the fuzz from kinetic data in the literature. 

These difficulties with the definition of reaction time lead to frequency factors that are difficult to predict or reconcile with physical realities on the basis of fundamental concepts that will be discussed in Chapter 9. The only solution is therefore to make sure that die definition of space time is clearly reported and that the units used can be changed by the reader to other units using clearly understood conversion factors. 

The size of the pre-exponential (frequency factor) of k will depend on the units of A. In turn A depends on the units of the rate as measured that is, on the concentration units and time units as defined in the space time. This is where the definitions of space time, discussed in Chapter 2, have their influence on the rate constant. 

The second definition requires a positive entropy, AS, of activation. In this view the more organized the transition state the less positive entropy is required for reaction. The universal frequency factor therefore represents maximum disorganization in the transition state. The rate is maximized in this view when disorder is maximized. 

These are two opposite interpretations of the entropy effects. The second definition requires a limit to the size of the entropy in order not to exceed the universal frequency factor. All in all, it is the first interpretation that is the most informative. 

Our treatment, based on both the collision and the statistical formulations of reaction rate theory, shows that there exist two possibilities for an interpretation of the experimental facts concerning the Arrhenius parameter K for unimolecular reactions. These possibilities correspond to either an adiabatic or a non-adiabatic separation of the overall rotation from the internal molecular motions. The adiabatic separability is accepted in the usual treatment of unimolecular reactions /136/ which rests on transition state theory. To all appearances this assumption is, however, not adequate to the real situation in most unimolecular reactions.The nonadiabatic separation of the reaction coordinate from the overall rotation presents a new, perhaps more reasonable approach to this problem which avoids all unnecessary assumptions concerning the definition of the activated complex and its properties. Thus, for instance, it yields in a simple way the rate equations (7.IV), corresponding to the "normal Arrhenius parameters (6.IV), which are both direct consequences of the general rate equation (2.IV). It also predicts deviations from the normal values of the apparent frequency factor K without any additional assumptions, such that the transition state (AB)" (if there is one) differs more or less from the initial state of the activated molecule (AB). ... 

Notation. The symbols a, f, y are used throughout to denote the electric field polarizability, first and second hyperpolarizabilities respectively, suitably qualified by frequency factors where necessary. The magnetizability is denoted by y and the nuclear screening tensor by a. The numerous but well-known acronyms specifying the computational procedures are used without definition. The possibly rather less well-known acronyms for the principal gauge invariant procedures are given in Table 1. 

Miscellaneous.— Isomerization of [Mo02(dipivaloylmethanato)2] is definitely intramolecular, but it is not possible to choose between a purely non-dissociative twist and a mechanism in which one or both of the bidentate ligands imdergoes partial dissociation to become unidentate. The activation energy for this isomerization is 17.0 kcal mol the frequency factor is gi2.6 454 Interconversion of isomers of octahedral molybdenum(v) complexes of the type MoOL LlX is fast. ... 

Once the reactor equations and assumptions have been established, and HDS, HDN, HDA, and HGO reaction rate expressions have been defined, the adsorption coefficient, equilibrium constants, reaction orders, frequency factors, and activation energies can be determined from the experimental data obtained at steady-state conditions by optimization with the Levenberg-Marquardt nonlinear regression algorithm. Using the values of parameters obtained from steady-state experiments, the dynamic TBR model was employed to redetermine the kinetic parameters that were considered as definitive values. The temperature dependencies of all the intrinsic reaction rate constants have been described by the Arrhenius law, which are shown in Table 7.4. 

The sinc fiinction describes the best possible case, with often a much stronger frequency dependence of power output delivered at the probe-head. (It should be noted here that other excitation schemes are possible such as adiabatic passage [9] and stochastic excitation [fO] but these are only infrequently applied.) The excitation/recording of the NMR signal is further complicated as the pulse is then fed into the probe circuit which itself has a frequency response. As a result, a broad line will not only experience non-unifonn irradiation but also the intensity detected per spin at different frequency offsets will depend on this probe response, which depends on the quality factor (0. The quality factor is a measure of the sharpness of the resonance of the probe circuit and one definition is the resonance frequency/haltwidth of the resonance response of the circuit (also = a L/R where L is the inductance and R is the probe resistance). Flence, the width of the frequency response decreases as Q increases so that, typically, for a 2 of 100, the haltwidth of the frequency response at 100 MFIz is about 1 MFIz. Flence, direct FT-piilse observation of broad spectral lines becomes impractical with pulse teclmiques for linewidths greater than 200 kFIz. For a great majority of... 

Adverse event. Unwanted effects that occur and are detected in populations. The term is used whether there is or is not any attribution to a medicine or other cause. Adverse events may be known parts of a disease that are observed to occur within a period of observation, and they may be analyzed to test for their frequency in a given population or trial. This is done to determine if there is an unexpectedly increased frequency resulting from nondisease factors such as medicine treatment. The term adverse event or adverse experience is used to encompass adverse reactions plus any injury, toxicity, or hypersensitivity that may be medicine-related, as well as any medical events that are apparently unrelated to medicine that occur during the study (e.g., surgery, illness, and trauma). See definition of Adverse reaction. 

Thus, the steric factor may be explained with the help of entropy change. When two molecules come together to produce the activated complex, the total translational degrees of freedom are reduced (from 6 to 3) and rotational degrees of freedom also diminish. This is compensated by an increase in vibrational degrees of freedom. But the definite orientation in forming the activated complex necessarily reduced the entropy, i.e. AS is negative. This decrease in entropy is small when reaction takes place between simple atoms. The calculated value of kbT/h corresponds to collision frequency... 

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