Lindemann’s theory

Another difficulty with Lindemann s theory becomes apparent when equation (4.59) is considered from another point of view. Equation (4.59) may be written as... 

It is thus necessary to modify Lindemann s theory to explain the deviations. These deviations have been explained by theories proposed by Hinshelwood, Kassel, Rice and Ramsperger, and Slater. 

In Hinshelwood s treatment, the molecule A is allowed to acquire an amount of energy El at an enhanced rate. The rate at which A converts to A is independent of that energy. Let us take the expression for first order rate constant given by Lindemann s theory, i.e. 

In most chemical reactions the rates are dominated by collisions of two species that may have the capability to react. Thus, most simple reactions are second-order. Other reactions are dominated by a loose bond-breaking step and thus are first-order. Most of these latter type reactions fall in the class of decomposition processes. Isomerization reactions are also found to be first-order. According to Lindemann s theory [1, 4] of first-order processes, first-order reactions occur as a result of a two-step process. This point will be discussed in a subsequent section. 

This reaction also involves the elimination of carbon monoxide and the formation of a mixture of hydrocarbons, principally ethane and methane. It is homogeneous and conveniently measurable between 450° and 600° C. The decomposition is kinetically unimolecular over a considerable range of pressure, but at pressures below about 80 mm. Hg the velocity constant falls appreciably, in the manner which would be expected if Lindemann s theory were correct. In the region of pressure where the reaction is unimolecular the velocity constants (sec-1) are given by... 

The chain theory can obviously provide a rate of activation great enough to account for any observed rate of reaction. With Lindemann s theory it is necessary that the normal rate of production of activated molecules by collision should be at least equal to and indeed considerably greater than the number of molecules undergoing chemical transformation in unit time. 

In Lindemann s theory of active intermediates, decomposition of the intermediate does not occur instantaneously after internal activation of the molecule rather, there is a time lag, although infinitesimally small, during which the species remains activated. For the azomethane reaction, the active intermediate is formed by the reaction... 

Summary.—The mechanism of the activation process in gaseous systems has been investigated from the point of view of (1) activation by radiation (2) activation by collision. An increase in the radiation density of possible activating frequencies has resulted in no increased reaction velocity. The study of the bimolecular decomposition of nitrous oxide at low pressures has led to the conclusion that the reaction is entirely heterogeneous at these pressures. A study of the unimolecular decomposition of nitrogen pentoxide between pressures of 7io mm. Hg and 2 X 10 3 mm. Hg shows no alteration in the rate of reaction such as was found by Hirst and Rideal but follows exactly the rate determined by Daniels and Johnson at high pressures. No diminution of the reaction velocity as might be ex-expected from Lindemann s theory was observed. 

With decrease in pressure, the chance that an activated molecule will lose its energy decreases more rapidly than the chance of the energy becoming so distributed as to allow a reaction to occur. Consequently, at low pressures the rates of unimolecular reactions cannot remain independent of the pressure. The activated molecule tends to become a Van t Hoff intermediate, and the rate to depend on the collision frequency. This corollary of Lindemann s theory has been... 

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). 

Lindemann s suggestion was amplified in 1926 by Hinshel-wood8 and by Fowler and Rideal.9 Independently and practically simultaneously Rice and Ramsperger10 proposed the fairly complete mathematical treatment for the collision theory which accounted for all the facts and predicted quantitatively the decrease in reaction velocity constant at decreased pressures. Kassel11 amplified these theories still further and introduced a refinement based on the quantum theory. 

More sophisticated treatments of Lindemann s scheme by Lindemann— Hinshelwood, Rice—Ramsperger—Kassel (RRK) and finally Rice— Ramsperger—Kassel—Marcus (RRKM) have essentially been aimed at re-interpreting rate coefficients of the Lindemann scheme. RRK(M) theories are extensively used for interpreting very-low-pressure pyrolysis experiments [62, 63]. 

Thermal unimolecular reactions usually exhibit first-order kinetics at high pressures. As pointed out originally by Lindemann [1], such behaviour is found because collisionally energised molecules require a finite time for decomposition at high pressures, collisional excitation and de-excitation are sufficiently rapid to maintain an equilibrium distribution of excited molecules. Rice and Ramsperger [2] and, independently, Kassel [3] (RRK), realised that a detailed theory must take account of the variation of decomposition rate of an excited molecule with its degree of internal excitation. Kassel s theory is still widely used and is valid for the chosen model of a set of coupled, classical, harmonic oscillators. 

The simple model outlined in the previous section would require that be a linear function of [M]". In fact, such plots of experimental data show marked curvature. The simple scheme fails because the mean time for decomposition of X decreases with its energy. In Kassel s theory [3], the Lindemann scheme is taken to be valid for a small energy range and ft, and fe3 are evaluated as a function of energy. 

All theories of unimolecular readtions are based upon Lindemann s mechanism, according to which energization and activation occur in two distinct stages, viz. 

Figure 2.13 is a sketch of the pressure dependence of a unimolecular reaction showing the two limiting conditions. The region joining the two extremes is known as the fall off region. Theories of unimolecular reactions have advanced considerably since Lindemann s initial proposal but they are still based on the same physical ideas so clearly highlighted in the Lindemann mechanism. 

For the reactions of medium-sized molecules we have the following Lindemann-Hinshelwood theory, RRK theory. Slater s harmonic theory, RRKM theory, phase-space theory, absolute reaction rate theory, quasi-equilibrium theory, and several others. All of those are grouped under the umbrella of "transition state theory" (Robinson Holbrook, 1972 Forst, 1973). Among these theories, some are regarded as "inaccurate" or "outdated." But several rivals remain as viable alternatives on which to base a theoretical study of a reaction system, at least as far as Joiunal referees are concerned. 

There is no statistical theory of the solid-liquid transition, even for hard spheres. So, to estimate volume fractions of coexisting phases, Marcelja et al. resorted to an intuitive approach based on Lindemann s empirical rule, which states that a solid will melt when the root mean-square displacement of particles about their equilibrium positions exceeds some characteristic fraction /l of the lattice spacing. For potentials of functional form r " (n >4) it is found that /l 0.10. For particles of charge zq occupying volume 4ira /3 on a f.c.c. lattice with spacing b, the Lindemann ratio is... 

This treatment is called the Lindemann-Hinshelwood theory. Although the treatment of Hinshelwood was successful in reproducing the experimental values of high-pressure limit rate constants k o, the theory stUl has one defect, that the values of s necessary to explain the experimental values differ largely from the acmal number of vibrational freedom and there is a large discrepancy of with experimental values in the fall-off region. 

In 1910 Frederick Lindemann proposed a simple idea solids melt when the amplitude of atomic thermal vibrations exceeds a fraction of the interatomic spacing. His quantitative model made use of Einstein s explanation of the low temperature specific heats of solids, which proposed that the atoms vibrate as quantized harmonic oscillators. Einstein s theory had been published only three years earlier, and its adoption by Lindemann appears to be the first application of quantum theory to condensed matter after Einstein s own paper. 

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]. 

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