Arrhenius expression, temperature coefficients

The temperature coefficient of the rate of a polymerization induced by a thermally decomposing initiator must depend according to Eq. (12) both on the temperature coefficient of kp/k] and on that of kd. Upon substituting Arrhenius expressions for each of the rate constants... 

It follows from this discussion that all of the transport properties can be derived in principle from the simple kinetic theory of gases, and their interrelationship through A, and c leads one to expect that they are all characterized by a relatively small temperature coefficient. The simple theory suggests that this should be a dependence on T1/2, but because of intermolecular forces, the experimental results usually indicate a larger temperature dependence even up to r3/2 for the case of molecular inter-diffusion. The Arrhenius equation which would involve an enthalpy of activation does not apply because no activated state is involved in the transport processes. If, however, the temperature dependence of these processes is fitted to such an expression as an algebraic approximation, then an activation enthalpy of a few kilojoules is observed. It will thus be found that when the kinetics of a gas-solid or liquid reaction depends upon the transport properties of the gas phase, the apparent activation enthalpy will be a few kilojoules only (less than 50 kJ). 

The reaction of hydrogen atoms with hydrazine has only been studied at temperatures between 25 and 200 °C53 54. At these temperatures the reaction proceeds via (5). This is confirmed by the observation that in the reaction D+N2H4 no NH2D could be detected this rules out reaction (4) at low temperatures. For the rate coefficient of reaction (5) Schiavello and Volpi54 quoted the Arrhenius expression... 

An Arrhenius expression, k = 1012 exp(—24,300/Hr) sec-1, was quoted for the coefficient of the rate-determining step. It is doubtful whether this step is a true unimolecular reaction. The effect of pressures has not been studied. The range of temperatures used appears to be too narrow to discuss the temperature dependence... 

The temperature dependence of the diffusion coefficients can be described by the conventional form of the Arrhenius expression,... 

At elevated temperatures, the organic liquid A isomerises smoothly to B, which is also a liquid, by an irreversible first-order reaction. The rate coefficient k is given by the Arrhenius expression... 

Choosing convenient values of Xa, calculating the temperatures from eqn. (38) and using the Arrhenius expression (36) to obtain the rate coefficient corresponding to each value of Xa, allows Table 1 to be drawn up. 

Obviously, reaction selectivity will depend on temperature. For isothermal operation, the temperature can be chosen to maximise the yield of desirable product. For example, consider the simultaneous parallel reactions (12). If the rate coefficients conform to the Arrhenius expression, e.g. 

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 ... 

The temperature coefficient of the reaction is still given by the Arrhenius equation. It is reasonable to assume that the velocity constant of the reaction is proportional to the radiation density. Now chemical heats of activation correspond to frequencies in the short infra-red region, and for these values of v the term ehvlkT in Planck s equation is large in comparison with unity. The expression for Uy thus reduces to... 

Activation energy was introduced in 1889 by -> Arrhenius in a paper [ii] which dealt with the temperature dependence of the - rate coefficient. The - Arrhenius expression (equation) in that a appears is valid over a finite temperature range. a is usually determined by plotting In k vs. l/T on the basis of the following expression [iii,iv]... 

While their results show considerable scatter (the apparent second-order rate coefficients are reproducible only to within a factor of 4) they indicate quite clearly the influence of added argon the on rate of NOCl decomposition thus the uni-molecular decomposition route is dominant. This result confirms the interpretation of Ashmore and Spencer mentioned previously. Extrapolation of the Arrhenius expression obtained at low temperatures fails by at least an order of magnitude to predict the high-temperature shock-tube results. 

These equations are commonly called pyrolysis relations, in reference to the thermal (as opposed to a possibly chemical or photonic) nature of the initiating step(s) in the condensed phase decomposition process. It can be seen that while the second, simpler pyrolysis expression with constant coefficient As) preserves the important Arrhenius exponential temperature dependent term, it ignores the effect of the initial temperature, condensed phase heat release and thermal radiation parameters present in the more comprehensive zero-order pyrolysis relation. These terms To, Qc, and qr) make a significant difference when it comes to sensitivity parameter and unsteady combustion considerations. It is also important to note the factor of 2, which relates the apparent "surface" activation energy Es to the actual "bulk" activation energy Ec, Es- E /1. Failure to recognize this factor of two hindered progress in some cases as attempts were... 

Contribution of a Secondary Reaction of Atomic Oxygen with Vibrationaiiy Excited CO2. Assuming a simple Arrhenius expression for the rate coefficient of the secondary reaction (5-7), compare the reaction rate with the three-body recombination of O atoms. Estimate the vibrational temperature at which the contributions of these two channels are comparable. 

The temperature coefficient E in the Arrhenius expression for a rate constant k = A exp (-E/RT) where A is temperature-independent and called the pre-exponential factor or frequency factor. 

If the polymer is above its glass transition tempetature, Tg, it responds rapidly to changes in its physical condition and we have Fickian or Case I diffusion. This is the simplest case, and for T>T, Henry s law is valid for sorption and the diff ion coefficient is a constant (ideal Fickian diffusion). Its temperature dependence is well approximated by a simple Arrhenius expression with a constant activation energy. 

Finally, by evaluating the derivative of (28) with respect to temperature, it is possible to derive a relationship between the above thermodynamic quantities and the empirical Arrhenius expression for reaction rate coefficients (15) ... 

Fig. 2.12. The points show the experimentally determined values of the rate coefficients for the reactions of 0( P) atoms with alkenes at different temperatures, whilst the dashed lines show the results of the theoretical calculations described in the text. The solid lines to the right of the diagram represent the Arrhenius expressions recommended by Cvetanovic to fit kinetic data between 300 and 700 K.

If we substitute Arrhenius expression for the diffusion coefficient (6) into Equation (23) and then perform some simple algebraic transformations, we may obtain the following temperature dependence of / / ... 

The tendency of solvent molecules to associate and to form an inner structure is decreased. The internal friction between molecules is reduced. This means that the temperature coefficient of viscosity is always negative, i.e. the viscosity is decreasing with increasing temperature. As a general rule, the viscosity decreases by ca. 2 % [36]. The temperature dependence for many liquids is expressed by a relationship of the Arrhenius type (Fig. 2.5) ... 

The temperature coefficient usually decreases with rise in temperature. The variation of velocity constant with temperature is best expressed by the following empirical equation proposed by Arrhenius ... 

Let us reconsider expression [7.54]. Coefficients K l, k2, and 4 are all rate coefficients that obey Arrhenius law with a temperature coefficient equal to the activation energy of the corresponding steps. The speed will not obey Arrhenius law. However, if the experimental results agree with each other, and with a straight line in coordinates (In v, 1/T), the calculated temperature coefficient will carry no physical meaning. 

We obtain a rate proportional to the concentration of A (first order) in the gas volume. In this expression, only constant k varies with temperature and therefore taking the logarithm of rate and expressing the temperature, we notice that Arrhenius law is followed and the temperature coefficient is equal to the activation energy (true or measured) of the reaction interface ... 

When rate coefficients are analyzed as a function of temperature, two-and three-parameter Arrhenius expressions can be obtained (Eqns (1) and (4)) as seen previously (Eqn (1)), the parameters from the latter are the data most commonly summarized in detailed mechanisms. In Arrhenius theory, the rate coefficient k T) is expressed as a function of activation energy ( ), preexponential factor A), temperature dependence factor ( ), the gas constant (R), and the temperature (T). 

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