Hinshelwood-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. 

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 ... 

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. 

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).

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 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. 

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. 

Figure 4. Proportional depletion by reaction with energy-dependent k2 E) and according to the Lindemann-Hinshelwood model (constant kz). The curve has been calculated at 999 K for ethane dissociation at the transition pressure.

In contrast, the Lindemann-Hinshelwood model assumes that all energized molecules react with the same rate constant k2- This model overestimates the contributions to uni from high energy states, and underestimates those from low-energy states. The true rate coefficient at some intermediate pressure will fall off faster than predicted by the Lindemann-Hinshelwood theory because of... 

The RRK theory has the virtue that it is very simple to apply and it does get close to the correct shape of the fall-off curve. As an example. Fig. 5 shows the fall-off curve calculated from classical RRK theory for the dissociation of cyclobutane using 14 oscillators. It can be seen that the theory is a considerable improvement on the Lindemann-Hinshelwood model. There are, however, some remaining problems. 

Troe proposed a similar approach to the calculation of the fall-off curve. In this case the zero-order approximation is the Lindemann-Hinshelwood model, formulated with the correct high and low-pressure limiting rate coefficients ... 

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

Equation (2.11) can be considered as a switching function describing the transition of k from feo to k with a value of k = k /2 at the center of the fall-off curve. We call Eq. (2.11), which is based on the model for unimolecular reactions that was first given 50 years ago by Lindemann and Hinshelwood, the Lindemann-Hinshelwood expression. It is shown in Fig. 2. 

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