Collision, diameter

Figure 7-12. Plot of the van der Waals interaction energy according to the Lennard-Jones potential given in Eq. (27) (Sj, = 2.0 kcal mol , / (, = 1.5 A). The calculated collision diameter tr is 1.34 A.

The Lennard-Jones 12-6 potential contains just two adjustable parameters the collision diameter a (the separation for which the energy is zero) and the well depth s. These parameters are graphically illustrated in Figure 4.34. The Lennard-Jones equation may also be expressed in terms of the separation at which the energy passes through a minimum, (also written f ). At this separation, the first derivative of the energy with respect to the internuclear distance is zero (i.e. dvjdr = 0), from which it can easily be shown that v = 2 / cr. We can thus also write the Lennard-Jones 12-6 potential function as follows ... 

A7 Ethane/methane selectivity calculated from grand canonical Monte Carlo simulations of mixtures in slit IS at a temperature of 296 K. The selectivity is defined as the ratio of the mole fractions in the pore to the ratio of mole fractions in the bulk. H is the slit width defined in terms of the methane collision diameter (Tch,- (Figure awn from Crackncll R F, D Nicholson and N Quirke 1994. A Grand Canonical Monte Carlo Study ofLennard-s Mixtures in Slit Pores 2 Mixtures of Two-Centre Ethane with Methane. Molecular Simulation 13 161-175.)... 

The collision diameter is at the value of s(r) equal to zero, and die maximum interaction of the molecules is where s(r) is a minimum. The interaction of molecules is thus a balance between a rapidly-varying repulsive interaction at small internuclear distances, and a more slowly varying attractive interaction as a function of r (Figure 3.7). 

Table 1. Values of Force Constants and Collision Diameters...

All of the examples of singlet energy transfer we have considered take place via the long-range resonance mechanism. When the oscillator strength of the acceptor is very small (for example, n-> n transitions) so that the Fdrster critical distance R0 approaches or is less than the collision diameter of the donor-acceptor pair, then all evidence indicates that the transfer takes place at a diffusion-controlled rate. Consequently, the transfer mechanism should involve exchange as well as Coulomb interaction. Good examples of this type of transfer have been provided by Dubois and co-workers.(47-49)... 

Consider a volume containing c A molecules of A (mass mA) and c B molecules of B (mass mB) per unit volume. A simple estimate of the frequency of A-B collisions can be obtained by assuming that the molecules are hard spheres with a finite size, and that, like billiard balls, a collision occurs if the center of the B molecule is within the collision diameter dAB of the center of A. This distance is the arithmetic mean of the two molecular diameters dA and dB ... 

Figure 6.9 (a) Collision diameter dAB (b) simplified basis for calculating frequency of A-B collisions... 

The intermolecular distance at which the reaction can occur may also be different from the collision diameter taken in the collision theory. 

Where V(r) is the pair interaction energy and Vm is the well depth. Here 2a is the collision diameter , i.e. the distance at which the interatomic potential is zero. Also plotted in Figure 2.2 is the interatomic force and the elastic modulus of what we may consider as an interatomic spring. (Recall that the force is the rate of change of energy with distance,... 

This volume fraction represents an increase in the effective particle size to give a collision diameter Reff... 

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