Hinshelwood-Lindemann theory

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. 

The second problem of the Lindemann-Hinshelwood theory is that the Lindemann plot is often far from a straight line, as can be seen by the example... 

Figure 2. Fall-off data for the dissociation of cyclobutane at 722 K [6,7], and comparison with Lindemann-Hinshelwood theory with 11 or 12 oscillators. The true number of vibrations is 30.

There is no reason for this expression to be consistent with the Lindemann straight line plot, but it is instructive to examine the physical reasons for the curvature. The low pressure limit is the same as in the Lindemann-Hinshelwood theory because the rate determining step is activation, which is dealt with in the same way in the two theories. This can be seen by taking the low pressure limit of Eq. (21). 

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... 

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 Among these theories, some are regarded as "inaccurate" or "outdated."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. 

Lindemann-Hinshelwood theory makes the assumption that a single collision with a bath gas molecule M is sufficient to deactivate AB to AB. In reality, each collision removes only a fraction of the energy. To account for the fact that not all collisions are fully deactivating, Troe (1983) developed a modification to the Lindemann-Hinshelwood rate expression. In the Troe theory, the right-hand side of... 

The LINDEMANN-HINSHELWOOD theory for unimolecular reactions is based on the following mechanism ... 

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. 

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. 

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 ... 

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]. 

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. 

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... 

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 ... 

If this Lindemann-Hinshelwood hypothesis is correct, unimolecular gas reactions should be first order at high pressures and should become second order at low pressures. This behavior has now been confirmed for a large number of reactions. In its original form the hypothesis had some difficulty in interpreting results quantitatively, but a number of extensions of the original hypothesis have been made, notably by R. A. Marcus whose treatment is consistent with transition-state theory. 

More detailed theories, as described in the following sections, take into account that in reality the rate coefficients of the Lindemann-Hinshelwood mechanism are averages over the quantum states of AB, AB, A, and B. For... 

Here log F ent gives the depression of the fall-off curve at the center relative to the Lindemann-Hinshelwood expression in a log k/k y sAog x) plot like Fig. 2. It turns out that F ent is a weak function of T and the nature of M that can be estimated by theory. Experimental fall-off curves can be fitted to the form of Eq. (2.13) and characterized by the three quantities ko, k, and F enf In essence, Eq. (2.13) is a first step beyond the Lindemann-Hinshelwood expression. Still more realistic, but more complex, expressions are given in Section 5. 

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. 

Note The Lindemann mechanism was also suggested independently by Christiansen. Hence, it is also sometimes referred to as the Lindemann-Christiansen mechanism. The theory of unimolecular reactions was further developed by Hinshelwood and refined by Rice, Rampsberger, Kassel and Marcus. 

Although the Lindemann theory is often satisfactory, it is incomplete since it does not fully recognise the relation between translational and internal energies. In many reactions the rate of activation by collision is not itself explicable unless it is assumed that activation can also occur by the transfer of vibrational energy from one molecule to another. This possibility was recognised by Hinshelwood and by Lewis and may be equivalent, in effect, to multiplying the frequency factor by 10" or more. 

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