Lindemann theory decomposition

M is Br2 or any other gas that is present. By the principle of microscopic reversibility , the reverse processes are also pressure-dependent. A related pressure effect occurs in unimolecular decompositions which are in their pressure-dependent regions (including unimolecular initiation processes in free radical reactions). According to the simple Lindemann theory the mechanism for the unimolecular decomposition of a species A is given by the following scheme (for more detailed theories see ref. 47b, p.283)... 

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... 

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. 

Conditions necessary for neglecting dc i/dt in the manner employed above may be investigated through formal approximations in reaction-rate theory. This will be considered further, with application to the Lindemann mechanism, in Section B.2.5. The mechanism itself generally contains fundamental inaccuracies and is best viewed as a simplified approximation to more-complex mechanisms. In particular, molecules capable of experiencing unimolecular decomposition or isomerization may exist in many different vibrationally excited states, and the rate constant for the reaction may differ in each state. Approximate means for summing over states to obtain average rate constants have been developed an introduction to these considerations maybe found in [3]. 

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... 

In the Lindemann-Hinshelwood theory the Lindemann expression for the uni-molecular rate constant, Eq. (9), is still assumed to be correct, but an improved activation rate coefficient is obtained from the Hinshelwood formulation. The shape of the fall-off curve should therefore still be the simple form predicted by Lindemann. Reference to Fig. 2 shows that, for the cyclobutane decomposition reaction, the change in the activation rate coefficient brings the theory much closer to the experimental results, particularly at low pressure. However, the shape of the fall-off curve is still not correct the Lindemann-Hinshelwood model predicts a faU-off region that is too narrow, the true fall-off is broader. 

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. 

Collision theory does not deal directly with unimolecular reactions but touches on the subject through the Lindemann mechanism. Once the molecule has been provided with sufficient energy by collision, the problem is to calculate the rate constant for the unimolecular decomposition,... 

Holbrook and Marsh showed that the gas-phase decomposition of ethyl chloride is apparently first order, but the rate constant at 521°C decreases with decreasing initial pressure (Table 1.12) [14]. How well can the decrease in rate constant be accounted for using the simple activation theory of Lindemann Determine this value of k o, and compare with the author s sec . ... 

We have calculated the addition channel rate constant using the RRKM approach to unimolecular reaction rate theory, as formulated by Troe ( ) to match RRKM results with a simpler computational approach. The pressure dependence of the addition reaction (1) can be simply decribed by a Lindemann-Hinshelwood mechanism, written most conveniently in the direction of decomposition of the stable adduct ... 

The coUision theory of gaseous reactions requires two molecules to collide, suggesting that such reactions should be second-order. Many decompositions, e.g., N2O5, appear to be first-order at sufficiently high pressures of the gas. However, some such reactions do appear to be second-order at low gas pressure. In 1922, Lindemann proposed an explanation of these observations. 

A two-step mechanism, first suggested by LINDEMANN /135/, is accepted in all contemporary theories of unimolecular reactions The first step is the formation of an "activated molecule by ineleastic bimolecular collisions, which supply it with an internal total energy amount over a critical value E, and the second step is the decomposition of the activated molecule. At high presure, there exist a thermal equilibriiam between the activated molecules (E>E ) and the normal molecules (E[Pg.230]

Historically, the theory of the termolecular reaction mentioned in the previous paragraph has been developed through the unimolecular reaction theory. This paragraph describes unimolecular decomposition reacticms in some detail. The chemical formula for the unimolecular decomposition reactions corresponding to the Lindemann mechanism can be shown as... 

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