Lindemann equation

As a good approximation it is assumed, that the adsorbed species are vibrating in resonance with the lattice phonon vibrations of the solid stationary phase. The phonon frequency can be evaluated from phonon spectra, from the standard entropy of solid metals, from the Debye temperatures or from the Lindemann equation [9]. 

The correct treatment of the mechanism (equation (A3.4.25), equation (A3.4.26) and equation (A3.4.27), which goes back to Lindemann [18] and Hinshelwood [19], also describes the pressure dependence of the effective rate constant in the low-pressure limit ([M] < [CHoNC], see section A3.4.8.2). 

The system of coupled differential equations that result from a compound reaction mechanism consists of several different (reversible) elementary steps. The kinetics are described by a system of coupled differential equations rather than a single rate law. This system can sometimes be decoupled by assuming that the concentrations of the intennediate species are small and quasi-stationary. The Lindemann mechanism of thermal unimolecular reactions [18,19] affords an instructive example for the application of such approximations. This mechanism is based on the idea that a molecule A has to pick up sufficient energy... 

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... 

There is, however, little exact data for its application. Lindemann and Magnus found that C, could be fairly well represented by adding to Einstein s equation an arbitrary term... 

That this is not the case follows from the experimental data discussed by A. Russell (9), and F. Koref (10) has attempted to calculate the change of frequency of an element when it enters into combination by means of the alteration of melting-point and atomic volume. According to Lindemann s equation, for the combined atom ... 

Since the Lindemann ratio dija 0.1 is empirically roughly the same for all substances, one expects the g value, as measured by sound attenuation, to be correlated with the glass transition temperature. Note that this relationship is independent of the details of the bead assignment. Equation (20), if rewritten as Tg di/a) g, is almost obvious, given the interaction of the form in Eq. (17) The typical lattice displacement, at Tg, is roughly A< ) j- dija. On the other hand, the typical structural excitations have the energy of about Tg, at the glass transition. 

If the stoichiometric equation for unimolecular reaction is A -> B + C, and if the energized molecules are denoted by A, the Lindemann mechanism consists of the following sequence of events. 

A test of the Lindemann mechanism is normally applied to observed apparent first-order kinetics for a reaction involving a single reactant, as in A - P. The test may be used in either a differential or an integral manner, most conveniently by using results obtained by varying the initial concentration, cAo (or partial pressure for a gas-phase reaction). In the differential test, from equations 6.4-20 and -20a, we obtain, for an initial concentration cAo = cM, corresponding to the initial rate rPo,... 

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... 

Conventional wisdom concerning thermal unimolecular reactions would seem to dictate that this must then be a Lindemann-type collisionally activated dissociation reaction scheme such as is in Equation (17). Application of the steady-state... 

As carried out above for the Lindemann mechanism, application of the steady-state approximation gives the apparent unimolecular rate constant in Equation (24) where [Av] represents the IR photon density. Again two limits may be considered. 

Our identification of the Buchenau relation in Eq. (64) with the AG equation for X further implies that the characteristic temperatures of glass formation (Tq, Tg, T, Ta) have their counterparts in characteristic values of (m ) and that the Lindemann criterion (see Section VI) can be generalized to describe successive stages of glass formation. Dividing the arguments of the exponentials defining... 

The last term F in this equation is equal to 1 for the Lindemann theory (it has been added for generality). 

The simplest type of system that obeys equation (17) is the unimolecular process 95li products. Since a stable molecule should not spontaneously break up into reaction products, the mechanism by which the unimolecular process occurs must be explained. Many unimolecular reactions are believed to follow the mechanism proposed by Lindemann, namely. 

This equation predicts that a graph of 1/ uni against 1/[M] (or /p) will be a straight line whose intercept is 1 /ho and whose slope is 1 /k. This type of plot is called a Lindemann plot, and provides, in principle, an elementary method for determining both the high and low-pressure limiting rate coefficients and for testing the theory. Unfortunately, as discussed below, Lindemann plots are often not linear. 

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