Lindemann model

The important feature to note about the Lindemann model is that it predicts that any spontaneous molecule reaction process can exhibit a first-order rate law at sufficiently high pressures and a second-order rate law at sufficiently low pressures. These features have now been well confirmed experimentally. 

The first problem discovered with the Lindemann model is the use of the simple collision theory, Eq. (1) for the activation rate constant. The collision frequency Zam can be calculated simply from collision theory, leaving the activation energy Eq to be determined. The low pressure limit is often difficult to attain experimentally, because collisions with impurities or collisions between... 

Both of these corrections to the Lindemann model are embraced by the... 

The Lindemann model discussed above provides the simplest framework for analyzing the dynamical effect of thermal relaxation on chemical reactions. We will see that similar reasoning applies to the more elaborate models discussed below, and that the resulting phenomenology is, to a large extent, qualitatively the same. In particular, the Transition State Theory (TST) of chemical reactions, discussed in the next section, is in fact a generalization of the fast thermal relaxation limit of the Lindemann model. 

The Hinshelwood-Lindemann model [5], in which molecules are activated and deactivated by collisions, is well accepted for describing the temperature T and pressure P dependence of thermal unimolecular reactions. The unimolecular rate constant. 

Although the Lindemann model can explain the occurrence of a fall-off region, it predicts its location to occur at several orders of magnitude higher pressures than experimentally observed. Related to this, calculated k2 rate constants from experimental high-pressure rate constants were found to be unrealistically large. One major cause of these problems is the inherent assumption that AB and AB° can be treated as different species. If so, then we obtain with AGa Eo... 

In general, the Hinshelwood-Lindemann model reproduces the location of the fall-off region well, though the shapes of experimental fall-off curves are still not accurately captured. To further improve the theory, a modihed reaction scheme proves to be helpful ... 

Simple fall-off reactions are reasonably well described by Linde-mann s collision activation model discussed in Section 2. It leads to an expression for the rate constant k T, p) based on Jcq, koo, and [M] (see equation (8)). Improved treatments of single-well single-channel reactions differ only in details from the Lindemann model. Therefore, it seems natural to base a general fall-off description on an extension of equation (8) ... 

The reaction rate coefficient may also depend on the pressure. A simple model is the Lindemann model [3], According to this model, a unimolecnlar decomposition is only possible if the energy in the molecule is sufficient to break the bond. Therefore, it is necessary that before the decomposition reaction occurs, energy needs to be added to the molecule by collision with other molecules M. So, the excited molecule A can be decomposed into products or can disable through a collision. 

In addition to the Lindemann model, there is the theory of unimolecular reactions. If the reaction rate of a unimolecular reaction is written as... 

This ensures the correct connection between the one-dimensional Kramers model in the regime of large friction and multidimensional imimolecular rate theory in that of low friction, where Kramers model is known to be incorrect as it is restricted to the energy diflfiision limit. For low damping, equation (A3.6.29) reduces to the Lindemann-Flinshelwood expression, while in the case of very large damping, it attains the Smoluchowski limit... 

In a review of the subject, Ubbelohde [3] points out that there is only a relatively small amount of data available concerning the properties of solids and also of the (product) liquids in the immediate vicinity of the melting point. In an early theory of melting, Lindemann [4] considered that when the amplitude of the vibrational displacements of the atoms of a particular solid increased with temperature to the point of attainment of a particular fraction (possibly 10%) of the lattice spacing, their mutual influences resulted in a loss of stability. The Lennard-Jones—Devonshire [5] theory considers the energy requirement for interchange of lattice constituents between occupation of site and interstitial positions. Subsequent developments of both these models, and, indeed, the numerous contributions in the field, are discussed in Ubbelohde s book [3]. 

Support for this work by 3M Company and permission to publish is gratefully acknowledged. We thank Dave Lindemann for the finite-element modeling and 3M s Division Engineering, especially Don Peacock, for all their work. Bob Maline of 3M s Specialty Materials Division was supervisor of pilot plant operations. 

Chapter 5, vapor pressure isotope effects are discussed. There, a very simple model for the condensed phase frequencies is used, the Einstein model, in which all the frequencies of a condensed phase are assumed to be the same. From this model, one can derive the same result for the relationship between vapor pressure isotope effect and zero-point energy of the oscillator as that derived by Lindemann. 

The Slater theory starts initially from the earlier treatments of Hinshel-wood and Lindemann. It provides a d5Tiamic representation of how a molecule with sufficient energy reaches the configuration of the activated complex. The model treats all oscillators as simple harmonic. 

In spite of the proper qualitative features of the Lindemann-Hinshelwood model, it does not correctly predict the much broader experimental fall-off behavior this is shown in Fig. 18, in which log(fe/fc ,) is plotted as a function of log(M = P/RT/Mj = Pc/RT). As evident from this figure, the actual rate at the center of fall-off (i.e., at PJ is depressed relative to the L-H model consequently, the transition of rate from low- to high-pressure limit occurs more gradually. 

Fig. 18. Comparison of the Lindemann-Hinshelwood model (dashed line) with experimental data (solid line).

The constraint in Eq. (38) that enables the direct computation of Tg is obtained by the extension of the Lindemann criterion to the softening transformation in glass-forming liquids [42, 56, 129, 130], and the details of this relation are explained in Section VI. Within the schematic model for glass formation (with specified e, Es, and monomer structure), all calculated thermodynamic properties depend only on temperamre T, on pressure P, and on molar mass Mmoi (which is proportional to the number M of united atom groups in single chains). The present section summarizes the calculations for To, Tg, Ti, and Ta as functions of M for a constant pressure of P = 0.101325 MPa (1 atm). 

Of course, in a thermal reaction, molecules of the reactant do not all have the same energy, and so application of RRKM theory to the evaluation of the overall unimolecular rate constant, k m, requires that one specify the distribution of energies. This distribution is usually derived from the Lindemann-Hinshelwood model, in which molecules A become activated to vibrationally and rotationally excited states A by collision with some other molecules in the system, M. In this picture, collisions between M and A are assumed to transfer energy in the other direction, that is, returning A to A ... 

The first of the shortcomings of the Lindemann theory—underestimating the excitation rate constant ke—was addressed by Hinshelwood [176]. His treatment showed that ke can be much larger than predicted by simple collision theory when the energy transfer into the internal (i.e., vibrational) degrees of freedom is taken into account. As we will see, some of the assumptions introduced in Hinshelwood s model are still overly simplistic. However, these assumptions allowed further analytical treatment of the problem in an era long before detailed numerical solution was possible. 

The Hinshelwood model thus corrects one of the major deficiencies in the Lindemann theory of unimolecular reactions. The greater excitation rate constant of Eq. 10.132 brings the predicted fall-off concentration [M]j/2 of Eq. 10.109 into much better accord with experiment. However, because of the many simplifying assumptions invoked in the Hinshelwood model, there are still a number of shortcomings. 

While the underlying mathematical optimization problem, also referred to as Steiner-Weber-problem or minisum problem, is one of the classical models discussed in operations research literature on facility location (cf. Drezner et al. 2001), it is much too abstract to be of real value to actual industrial location decisions (cf. Gotze 1995, p. 56). A general criticism of Weber s theory can be found in Behrens (1971, pp. 15-19) and Meyer-Lindemann (1951, pp. 55-67). 

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