Line of centers model

Therefore, Eact = RT, which shows that even when threshold energy E() is zero, the Eact has a non-zero value. In case of line of center model, a collision is reactive only if the component of the relative translational energy along the line joining the centre of mass of the two molecules exceeds E0. In case of line of centers of model, the rate constant is given by... 

This section considers the cross section for reactive collisions ar. Bimolecular reactions will be treated explicitly. The rate (frequency) of collisions depends on the collision cross section. The larger the cross section, the more often molecules run into one another. In a similar way the reactive cross section determines how often molecules run into one another and react. This section introduces the simple line-of-centers model for scaling of the reactive cross section with energy. 

Suppose that there is an energetic barrier e that must be overcome for a reaction to occur, for example, the energy needed to break a critical chemical bond. The translational energy of the relative velocity of the collision partners is available to surmount the reaction energy barrier. We consider a simple picture called the line-of-centers model of reactive collisions. In this model only the velocity directed along the line-of-centers between the two molecules at the point of collision is effective in overcoming the barrier to reaction. 

In the line-of-centers model, the collision s effectiveness depends on the impact parameter, b. In general, the reaction probability is a function of energy and impact parameter, P (b, e). An integral over all possible impact parameters... 

Janssen, M.H.M. and Stolte, S. (1987) Calculation of steric effects in reactive collisions employing the angle-dependent line of centers model. J. Phys. Chem. 91, 5480-5486. 

Connor, J.N.L., Whitehead, J.C. and Jakiibetz, W (1987) Orientation dependence of the F + H2 reaction - analysis of the angle-dependent line-of-centers model. J. Chem. Soc., Faraday Tram. 83. 1703-1718. 

The preexponential factor in the line-of-centers model is proportional to which is a direct consequence of the cross-section function (8.13). With a different, but less readily rationalized, model for 0(e), a constant preexponential factor is predicted (Problem 8.18). 

Therefore the reaction criterion (3.29) is sometimes formulated as reaction occurs if the kinetic energy along the line of centers of two hard spheres exceeds Eo- The criterion and the cross-section (3.30) is then known as the line of centers model. This interpretation of (3.29) is correct but not essential. We can think of Eo as the value of V(R) atR = d. Equation (3.29) is then the criterion that the motion of the reactants under the potential V(R) can reach the separation R = d where V(d) = Eo and Eo is positive. 

Figure 3.11 Orientation dependence of the cross-section for the reaction H + D2(v= /= 0) HD + D at the two indicated values of the collision energy Ej- The ordinate is dff R/dcos y = 2ctr(cos y). The solid curves were calculated from the angle-dependent line-of-centers model, Eq. (3.34), and the (open and filled) points represent dynamical computations (these are quasi-classical trajectory results that have statistical error bars as discussed in Chapter 5) on the ab initio potential surface referred to in Figure 3.10 [adapted from N. C. Blais, R. B. Bernstein, and R. D. Levine, J. Phys. Chem. 89, 20 (1985)].

M. The line-of-centers model with a steric requirement, (a) Derive the result (3.35) for the energy dependence of the reaction cross-section when the barrier depends on the approach angle. The point is that you will have to be care fid about the limits of integration and this will make you think in detail about the energy dependence of the cone of acceptance. See Smith (1980). (b) Derive the results... 

Hence, E increases with temperature for the line-of-centers model (cf. Fig. 2). 

Figure 6. Schematic representation of specific Fj, Fq (a) and effective (b) reaction functions for A + BC(i = 0,1) AB 4- C. The excitation functions Oi were assumed to have the same shape, corresponding to the line-of-centers model, but different threshold energies. The ratio of specific rate coefficients k jk = J Fj dF/J Fq dE was chosen to be approximately 20 at 300 K.

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