Modified Arrhenius equation

Equilibrium constants, determined in the laboratory or calculated from free energy data, are usually expressed by an Arrhenius equation, modified to take into account the effect of the difference in molar volumes between reactant and product solids (the C term, Eugster and Wones, 1962) ... 

Innumerable experimental rate measurements of many kinds have been shown to obey the Arrhenius equation (18) or the modified form [k = A T exp (—E/RT)] and, irrespective of any physical significance of the parameters A and E, the approach is an important, established method of reporting and comparing kinetic data. There are, however, grounds for a critical reconsideration for both the methods of application and the theoretical interpretations of observed obedience of experimental data for the reactions of solids to eqn. (18). 

A linear regression of A (ul, x) for < 0.8 was performed for each value of x, the slope giving A °(x) which is found to be linear in x. From the linear dependence of A °(x) on x, the original relaxation curves can be retraced. Inserting Eq. (86) into the modified Arrhenius equation ... 

This expression was used in the parameter estimation, i.e., just the concentrations cp and cmb were used in the data fitting, and Ca was calculated from Eq. (17). The rate constants included in the model were described by the modified Arrhenius equation... 

A rough estimate of the viscosity of a pure liquid at its boiling point can be obtained from the modified Arrhenius equation ... 

In these circumstances a modified approach to the shift factor can be used and the formal WLF equation abandoned. The shift factor concept is still used but for these situations an Arrhenius equation is fitted to the shift factor aT (not to log aT) ... 

These equations predict a continuously diminishing rate of creep. Many empirical and semi-empirical models of creep-strain have been made and are described by Ward [24], One of these has been used successfully to describe the later stages of creep in polymers such as oriented polyethylene. The Arrhenius equation was modified by Eyring to apply to the rate of creep (deJdt) in the following way ... 

When both temperature and pressure are considered, the modified Arrhenius equation is (Harrison et al., 1985)... 

Equation (3.35) is a precursor to the Arrhenius equation relating the rate constant [k in Eq. (3.30)] to the probability of molecular collisions (to) and the activated energy (Ea) of a reaction. Polysaccharide viscous flow is characterized by a modified Arrhenius equation in which T j/i 0 replaces k ... 

The model presented for a thermal explosion predicts that for a reaction mixture of fixed composition and fixed initial temperature, there will be a critical pressure above which explosion will occur and below which a normal stationary reaction will take place. The relation between the critical pressure and temperature is given by a modified Arrhenius equation with a negative temperature coefficient [Eq. (XIV.3.8)] which is... 

In summary, it was observed that the alumina catalyst increased the extent of pitch pyrolysis at long reaction times, accompanied by some changes in the product distribution. The pyrolysis reaction was found to be first order with respect to the pitch reactant. A compensation effect was observed in which the Arrhenius parameters Ea and A were found to vary with the residue weight, WR. This effect was explained in terms of a modified Arrhenius equation and of the increasing activation energy Ea associated with the less reactive molecular species. 

It was found that the temperature data for reactions (10) and (18) could be represented by modified Arrhenius equations with identical pre-exponential factors. In its thermodynamic formulation TST defines a bi-molecular rate coefficient as... 

The Arrhenius equation and the energy balance equation were used to obtain a modified version of the equation for the second order adiabatic reaction kinetics. 

Fragmentations are unimolecular reactions, and thus, they follow the Arrhenius law (Eq. (5.9)). Since the actual temperature is unknown, the Arrhenius plot of In fe over l/T needs to be modified in that In k is plotted against In P. Linear relationsships are obtained. The activation energy Ea can be determined from the slope of the line. Ea contains the enthalpic contributions to the barrier. The entropic contributions remain unknown, because they are included in the preexponential factor A of the Arrhenius equation. This factor can only be determined from the intersection of the line with the ordinate and since the actual temperature is not known, there is no way to determine the intersection and the pre-exponential factor. 

The Tafel equation rj = a b ni, where fc, the so-called Tafel slope, conventionally written in the form b = RT/aF, where a is a charge transfer coefficient, has formed the basis of empirical and theoretical representations of the potential dependence of electrochemical reaction rates, in fact since the time of Tafel s own work. It will be useful to recall here, at the outset, that the conventional representation of the Tafel slope as RT/aF arises in a simple way from the supposition that the free energy of activation AG becomes modified in an electrochemical reaction by some fraction, 0.5, of the applied potential expressed as a relative electrical energy change rjF, and that the resulting combination of AG and 0.5tjF are subject to a Boltzmann distribution in an electrochemical Arrhenius equation involving an exponent n I/RT. Hence we have the conventional role of T in b = RT/aF, as will be discussed in more detail later. 

One additional kinetic theory that deserves to be mentioned is the theory based on the Eyring equation. The essence in this theory is captured by the following phrase credited to Henry Eyring in 1945 "... a molecular system. .. passes. .. from one state of equilibrium to another. .. by means of all possible intermediate paths, but the path that is most economical in energy will be more often traveled." The equation of the chemical reaction (6.19), which was used for the interpretation of the kinetic theory based on the Arrhenius equation, is modified in order to explain the Eyring equation, as shown in equation (6.20). 

Use a modified form of the Arrhenius equation to calculate the temperature at which the rate constant is... 

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). 

In this Section, an approach to the descnption of kinetics of deformation, relaxation, and fracture of polymers over a wide temperature range will be elaborated on the basis of kmetic equations with constant activation parameters. A nonuniform distribution of the energy of atoms over the degrees of freedom requires the replacement of T by the F(6/T) quantum function. The resultant (modified) Arrhenius equation describes adequately both deformation and fracture of polymers in the range from 20 K to the melting temperature. 

The HX rates are also dependent on temperature. An increase in temperature affects HX rates primarily hy altering the water ionization constant, K, and thus increasing the concentration of OH . Further, some evidence suggests that temperature may also affect the collisional rate constant, k, in Equation 1.2 hy altering buffer viscosity and thus the diffusional collisional rate constant [24, 25]. A more recent study, however, has indicated that the effect of bulk viscosity on HX is negligible [30]. Theoretical HX rates can be determined as a function of temperature by a modified form of the Arrhenius equation (Eq. 1.4) and reference HX rate constants determined experimentally at 20°C ... 

What if the rate law is non-Arrhenius If the infinite pressure rate can be expressed with confidence in some other simple form, e.g. the modified Arrhenius equation... 

Since the temperature variation of RRKM rate constants is modified Arrhenius, equation... 

The 2,0 method assumes that the ratio of the times to equivalent damage at low temperatures (usually 10°C apart) has a constant value. In fact, the 2io v iU decrease with increasing temperature. Donohue and Apostolou suggest the use of a modified method, the variable 2io method, in which the ratio of the times to equivalent damage for two temperatures is used as a variable. In this method the TED ratio is equal to the Arrhenius equation, and the 2io is determined with the TED ratio as a variable as follows ... 

Strategy A modified form of the Arrhenius equation relates two rate constants at two different temperatures [see Equation (13.14)]. Make sure the units of R and are consistent. 

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