Lindemann plot

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. 

The Lindemann plot described above is typically not a straight line. 

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 3. Lindemann plot for the dissociation of cyclobutane at 722 K, illustrating deviation from linearity.

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. 

In the preceding expression, log(FJ is related to the depression of the fall-off curve at the center relative to the L-H expression in a og k/k ) vs. log(2f/(l -I- X)) plot. The values for F<. can then be related to the properties of specific species and reaction and temperature using methods discussed in Gardiner and Troe (1984). In Fig. 19, values of F for a variety of hydrocarbon decompositions are presented. As evident from this figure, in the limit of zero or infinite temperatures and pressures, all reactions exhibit Lindemann-Hinshelwood behavior and F approaches unity. From this figure, it is clear that L-H analysis generally does an adequate job in... 

Thus the Lindemann theory predicts that a plot of l/kUni versus 1 /[M] should yield a straight line. Experimental data consistently shows downward curvature at high pressure (small values of 1/ [M]) in plots of this type. The predicted v-intercept (1 /kam) is too large (i.e., the theory underpredicts the extrapolated infinite-pressure rate constant kUni,oo)- This is the second general breakdown of the Lindemann theory that motivated further theoretical analysis. 

Inclusion of the remaining three values drastically alters the intepretation. A grossly non-linear plot results, clearly illustrating the inadequacy of the simple Lindemann scheme. 

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. 

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). 

We check whether the cluster is solid by computing a modification of the Lindemann index [43] and by plotting the radial distributions of the molecular... 

Maier [170] and Lindemann et al. [171] have carried out cross-section measurements of reaction (107) as a function of ion kinetic energy. Maier used a longitudinal tandem mass spectrometer and Lindemann et al. a perpendicular tandem mass spectrometer. Thus the ion kinetic energies were rather well defined in both of the studies. C ions were produced by electron bombardment of CO under such conditions that all projectiles were in the ground state. Their results are summarized in Fig. 19 in which cross-sections are plotted versus barycentric energy. 

These points are plotted in Fig. 22.4. There are marked deviations at low pressures, indicating that the Lindemann theory is deficient in that region. 

Plot a series of five strict Lindemann curves, all with the same value of k, but with... 

Figure 14.21 Plot of the pressure dependence of the rate for a unimolecular decomposition that follows the Lindemann mechanism.

In this case, the rate law is first order in A and independent of the total pressure. A plot of the pressure dependence of a unimolecular decay that obeys the Lindemann mechanism is shown in Figure 14.21. This dependence is consistent with that which is experimentally observed in many unimolecular decompositions in the gas phase. 

In Fig. 5.5, log(A ex/ oo) is plotted for the decomposition of cyclopropane. While it appears to be a linear function of log [A] at low pressure, suggesting that the low-pressure limit has been reached, the slope is closer to than 1, the value predicted by (5.21). Such differences are typical the actual falloff with pressure is more gradual than is consistent with the simple Lindemann mechanism. Improvement requires considering the specific... 

The second difficulty with the Lindemann-Christiansen mechanism becomes apparent when experimental data are plotted in another way... 

For the Lindemann approach the rates k and k would be independent of energy. In terms of ideal gas law one would have [D]/[S] = d// s[M] = RTkjk P and one should expect a linear relation of [D]/[S] versus P. This is not verified the plot has a pronounced downward curvature. The rate of decomposition should increase with an increase in the vibrational energy. 

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. 

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