Arrhenius equation collision model

Equation 21 has an exponential temperature dependence, which is much stronger than the weak temperature dependence of the collision rate itself. Let s now confirm that our model is consistent with the Arrhenius equation. When we take logarithms of both sides, we obtain... 

This expression has exactly the same form as the Arrhenius equation, so our model is consistent with observation. Moreover, we can now identify the term orNA2 as the pre-exponential factor A and Emin as the activation energy, Ea. That is, A is a measure of the rate at which molecules collide, and the activation energy Ea is the minimum kinetic energy required for a collision to result in reaction. 

Equation (15.10) is a linear equation of the type y = mx + b, where y = ln( ), m = —EJR = slope, x = 1/T, and b = ln(A) = intercept. Thus, for a reaction where the rate constant obeys the Arrhenius equation, a plot of n(k) versus 1/T gives a straight line. The slope and intercept can be used to determine the values of a and A characteristic of that reaction. The fact that most rate constants obey the Arrhenius equation to a good approximation indicates that the collision model for chemical reactions is physically reasonable. 

Both the Arrhenius and the Eyring equation describe the temperature dependence of reaction rate. Strictly speaking, the Arrhenius equation can be applied only to gas reactions. The Eyring equation is used in the smdy of gas, condensed and mixed phase reactions - aU places where the simple collision model is not very helpful. The Arrhenius equation is founded on the empirical observation that conducting a reaction at a higher temperature increases the reaction rate. The Eyring equation is a theoretical construct, based on transition state model. 

The Arrhenius equation was developed empirically from the observations of many reactions. The two major models that explain the observed effects of concentration and temperature on reaction rate highlight different aspects of the reaction process but are completely compatible. Collision theory views the reaction rate as the result of particles colliding with a certain frequency and minimum energy. Transition state theory offers a close-up view of how the energy of a collision converts reactant to product. 

Thus, at least qualitatively, a hard-sphere collision model results in an equation which has a firm basis in experiment. This suggests that the Arrhenius parameters, which were treated as purely experimental parameters in Section 6, can be given a physical interpretation, at least for bimolecular processes. 

According to Equation 6.3, this factor is equivalent to the Arrhenius A-factor. In the collision model it is a measure of the standard rate at which reactant species collide that is it is a measure of the number of collisions per second when the concentrations of the reactant species are both 1 mol dm"-. It is necessary to specify standard conditions since, in general, the collision rate depends on the concentrations of the species present (cf. Section 4.1). The value of Atheory a given bimolecular reaction depends on the hard-sphere radii and masses of the reactant species. Calculations show that it does not vary significantly from reaction to reaction with values usually of the order of 10 dm mol s . Table 7.1 compares the calculated values of Atheory for gas-phase bimolecular reactions with those derived from experiment. 

Por the computation we have used the integral method using cubic spline and the combined gradient method of Levenberg-Marquardt [57, 58]. The kinetic models chosen describe well the hydrogenation kinetics. In the formulas presented in Table 3.1 k is the kinetic parameter of the reaction and Q takes into account the coordination (adsorption) of the product (LN) and substrate (DHL) with the catalyst (the ratio of the adsorption-desoprtion equilibrium constants for LN and DHL). Parameters of the Arrhenius equation, apparent activation energy kj mol , and frequency factor k, have been determined from the data on activities at different temperatures. The frequency factor is derived from the ordinate intercept of the Arrhenius dependence and provides a measure of the number of collisions or active centers on the surface of catalytic nanoparticles. 

The model uses three parameters that have to be fitted to the data K - a concentration distribution coefficient between the outermost layer and the first inner layer, C - the preexponential constant in an Arrhenius-type rate equation, and Ti - the efficiency of collisions with the external surface. The model contains additional parameters that can be determined from the characteristics of the adsorbent pellet and the operating conditions. 

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