Frequency factors, rate equation parameters

Lasocka parameter, activation energy of crystallisation, frequency factor, and Avrami parameter can be evaluated for different systems. Along with these parameters, crystallisation constants as a function of different temperatures as well as different heating rates (p) should be studied. Tg and its dependence on heating rates can be given by the following original empirical Lasocka equation. 

Activation Parameters. Thermal processes are commonly used to break labile initiator bonds in order to form radicals. The amount of thermal energy necessary varies with the environment, but absolute temperature, T, is usually the dominant factor. The energy barrier, the minimum amount of energy that must be suppHed, is called the activation energy, E. A third important factor, known as the frequency factor, is a measure of bond motion freedom (translational, rotational, and vibrational) in the activated complex or transition state. The relationships of yi, E and T to the initiator decomposition rate (kJ) are expressed by the Arrhenius first-order rate equation (eq. 16) where R is the gas constant, and and E are known as the activation parameters. 

The Arrhenius equation relates the rate constant k of an elementary reaction to the absolute temperature T R is the gas constant. The parameter is the activation energy, with dimensions of energy per mole, and A is the preexponential factor, which has the units of k. If A is a first-order rate constant, A has the units seconds, so it is sometimes called the frequency factor. 

If a data set containing k T) pairs is fitted to this equation, the values of these two parameters are obtained. They are A, the pre-exponential factor (less desirably called the frequency factor), and Ea, the Arrhenius activation energy or sometimes simply the activation energy. Both A and Ea are usually assumed to be temperature-independent in most instances, this approximation proves to be a very good one, at least over a modest temperature range. The second equation used to express the temperature dependence of a rate constant results from transition state theory (TST). Its form is... 

Another problem which can appear in the search for the minimum is intercorrelation of some model parameters. For example, such a correlation usually exists between the frequency factor (pre-exponential factor) and the activation energy (argument in the exponent) in the Arrhenius equation or between rate constant (appears in the numerator) and adsorption equilibrium constants (appear in the denominator) in Langmuir-Hinshelwood kinetic expressions. 

For fitting such a set of existing data, a much more reasonable approach has been used (P2). For the naphthalene oxidation system, major reactants and products are symbolized in Table III. In this table, letters in bold type represent species for which data were used in estimating the frequency factors and activation energies contained in the body of the table. Note that the rate equations have been reparameterized (Section III,B) to allow a better estimation of the two parameters. For the first entry of the table, then, a model involving only the first-order decomposition of naphthalene to phthalic anhydride and naphthoquinone was assumed. The parameter estimates obtained by a nonlinear-least-squares fit of these data, are seen to be relatively precise when compared to the standard errors of these estimates, s0. The residual mean square, using these best parameter estimates, is contained in the last column of the table. This quantity should estimate the variance of the experimental error if the model adequately fits the data (Section IV). The remainder of Table III, then, presents similar results for increasingly complex models, each of which entails several first-order decompositions. 

The temperature dependence of the rate constants of radical addition (k ) is described by the Arrhenius equation (Section 10.2). At a given temperature, rate variations due to the effects of radical and substrate substituents are due to differences in the Arrhenius parameters, the frequency factor, A , and activation energy for addition, . For polyatomic radicals, A values span a narrow range of one to two orders of magnitude [6.5 < log (A /dm3 mol-1 s-1) < 8.5] [2], which implies that large variations in fcj are mainly due to variations in the activation energies, E. This is illustrated by the rate constants and Arrhenius parameters for the addition to ethene of methyl and halogen-substituted methyl radicals shown in Table 10.1. 

The design parameters for a batch reactor can be as simple as concentration and time for isothermal systems. The number of parameters increases with each additional complication in the reactor. For example, an additional reactant requires measurement of a second concentration, a second phase adds parameters, and variation of the reaction rate with temperature requires additional descriptors a frequency factor and an activation energy. These values can be related to the reactor volume by the equations in Section III. 

Schwab has pointed out that the following relationship between the two parameters of the Arrhenius equation is frequently encountered. A decrease in the activation energy of a given reaction, for a series of catalysts, often does not increase the reaction rate to the extent calculated, because of a simultaneous decrease of the frequency factor. Cremer (106) confirmed this for the decomposition of ethyl chloride on various chloride catalysts. These findings will be discussed here with due regard to the relation between adsorption and elementary reaction rates dealt with in the preceding section. 

This expression relates the second-order rate constant, k, for an outer-sphere electron transfer reaction to the free energy of reaction, AG°, with one adjustable parameter, X, known as the reorganization energy. Wis the electrostatic work term for the coulombic interaction of the two reactants, which can be calculated from the collision distance, the dielectric constant, and a factor describing the influence of ionic strength. If one of the reactants is uncharged, Wis zero. In exact calculations, AG should be corrected for electrostatic work. The other terms in equation 46 can be treated as constants (Eberson, 1987) the diffusion-limited reaction rate constant, k, can be taken to be 10 M" is the equilibrium constant for precursor complex formation and Z is the universal collision frequency factor (see Eberson, 1987). 

The temperature dependence of the rate of reactions is particularly Important for the pyrolytic processes. Relation (5) can be used for the understanding of the common choices for the pyrolysis parameters. As an example, we can take the pyrolysis of cellulose [8]. Assuming a kinetics of the first order (pseudo first order), the activation energy in Arrhenius equation was estimated E = 100.7 kJ / mol. The frequency factor was estimated 9.60 10 s These values will lead to the following expression for the weight variation of a cellulose sample during pyrolysis ... 

The number of remaining parameters in (27) amounts to 8 the molecular mass of the coke, Mg, 2 frequency factors and 2 activa tion energies, 3 adsorption equilibrium constants, considered to be independent of temperature They were determined by non linear regression. F-tests were applied to the fit of the equation to the experimental data and t-tests to the parameter values The fit was excellent at 823 K, but less satisfactory at 873 K It could be improved and plausible values for the physico-chemical parameters could be obtained if two stages were considered in the rate of growth of coke instantaneous up to a certain intermediate size and finite beyond. 

Van t Hoff, as well as some other scientists, studied the increase in rate constants with increasing temperature. An earlier equation was modified by the Swedish chemist Svante Arrhenius to the form noted below. The Arrhenius equation is more than a semi-empirical equation to account for the usual doubling or tripling of reaction rate for every 18°F (10°C) increase. The E denotes the energy needed to induce reaction and A represents a frequency factor related to the probability of reaction. These parameters would be better understood during the 1930s with the development of transition-state theory. Wilhelm Ostwald s contributions to kinetics were many and included the application of thermodynamics to kinetics and mechanism as well as the explanation of catalysis. This magnificent triumvirate of physical chemists would all win Nobel Prizes in chemistry van t Hoff (1901) Arrhenius (1903) Ostwald (1909). 

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). ... 

Pyrolysis is a first order reaction so the temperature function of the reaction rate constant k and the half life time may be computed easily using the coefficients of the Arrhenius equation activation energy E and frequency factor A which had already been determined (see chapter 3.3.1, equations 3-7 and 3-8). Such data are the basis for the parameters of thermal conversion processes, such as temperature of the plant installations, housing time etc. 

The DSC experiments were carried out using a DuPont 990 Thermal Analyzer equipped with a 910 DSC and pressiue ceU. The experiments and their evaluation were carried out according to ASTM E 698-79 [3-13] using at least three different heating rates in a hydrocarbon atmosphere (CH ) at 10 bar pressure and with 5 cmVmin gas flow rate in order to simulate the process parameters of thermal cracking. As usual the coefficients of the Arrhenius equation (Eq. 3-37) were calculated, i.e. the activation energy E (kJ/Mole) and the frequency factor log A (min" ). 

Here is the activation energy, R is the universal gas constant, T is the temperature (in kelvins), and is a proportionality constant called the frequency factor or preexponential factor. We noted earlier that a large activation energy should hinder a reaction. Equation 11.9 shows this effect As increases, k will be smaller, and smaller rate constants correspond to slower reactions. Note that because the temperature and activation energy appear in the exponent, the rate constant will be very sensitive to these parameters. That s why fairly small changes in temperature can have drastic effects on the rate of a reaction. 

At the high recycling ratios the loop reactor operates as an ideal stirred-tank reactor. Therefore, the reaction rate can immediately be determined from the difference in concentration between the feed and the outlet, the throughput and the quantity of catalyst.The rate equation, describing the consumption of xylene and the formation of the reaction products, are considered to be pseudo first order. The parameter of the rate equations, which are the frequency factors and the activation energies, are determined by least square methods. In the above function (Fig. 6b) r is the measured rate, r is calculated with estimated parameters, w represent appropriate weight factors and N is the number of measured values. Because the rate equations could be differentiated v/ith respect to the unknown kinetic parameters, the objective function was minimized by a step-wise regression. 

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. 

Of the adjustable parameters in the Eyring viscosity equation, kj is the most important. In Sec. 2.4 we discussed the desirability of having some sort of natural rate compared to which rates of shear could be described as large or small. This natural standard is provided by kj. The parameter kj entered our theory as the factor which described the frequency with which molecules passed from one equilibrium position to another in a flowing liquid. At this point we will find it more convenient to talk in terms of the period of this vibration rather than its frequency. We shall use r to symbolize this period and define it as the reciprocal of kj. In addition, we shall refer to this characteristic period as the relaxation time for the polymer. As its name implies, r measures the time over which the system relieves the applied stress by the relative slippage of the molecules past one another. In summary. 

---