Lennard-Jones potential 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 molecular size of a solvent can be characterized in several ways. One of them is to assign the solvent a molecular diameter, as if its molecules were spherical. From a different aspect, this diameter characterizes the cavity occupied by a solvent molecule in the liquid solvent. From a still further aspect, this is the mean distance between the centers of mass of two adjacent molecules in the liquid. The diameter plays a role in many theories pertaining to the liquid state, not least to those treating solvent molecules as hard spheres, such as the scaled particle theory (SPT, see below). Similar quantities are the collision diameters a of gaseous molecules of the solvent, or the distance characterizing the minimum in the potential energy curve for the interaction of two solvent molecules. The latter quantity may be described, e.g., according to the Lennard-Jones potential (Marcus 1977)... 

Notes (Tlj = molecular collision diameter calculated from the Lennard-Jones potential ... 

The collision diameter o g is determined from the Lennard-Jones potential parameters for the two molecules... 

One-dimensional velocity distribution Specific conductivity, hard-sphere diameter for a collision, length parameter in Lennard-Jones potential, symmetry number of a molecule Intrinsic lifetime of a photoexcited state Azimuthal angular velocity in spherical polar coordinates, azimuthal angle in spherical polar coordinates, angle of deflection Quantum yield at wavelength A Fluorescence (phosphorescence) quantum efficiency... 

Lennard-Jones potential parameters such as collision diameter. Gay and energy of interaction, e y for nonpolar and nonreacting molecule pairs can be computed from the corresponding values of pure component species based on the following rules ... 

The ai is a characteristic diameter of the molecules of type 1, often called the collision diameter, and may be defined differently for each molecular model. In the Lennard-Jones potential has the significance of being the distance from the center... 

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 ... 

The second generalization is the reinterpretation of the excluded volume per particle V(). Realizing that only binary collisions are likely in a low-density gas, van der Waals suggested the value Ina /I for hard spheres of diameter a and for particles which were modeled as hard spheres with attractive tails. Thus, for the Lennard-Jones fluid where the pair potential actually is... 

In reality, molecules each occupy some space, so the empty volume of the container decreases as the concentration N/ V increases. In addition, there is generally some attraction even at distances substantially larger than the nominal diameter of the molecules, and the repulsive part is somewhat soft so that collisions are not instantaneous. The exact form of this interaction must be calculated by quantum mechanics, and it depends on a number of atomic and molecular properties as discussed in Chapter 3. For neutral, nonpolar molecules, a convenient approximate potential is the Lennard-Jones 6-12 potential, discussed in Chapter 3 Table 3.5 listed parameters for some common atoms and molecules. 

As already mentioned the present treatment attempts to clarify the connection between the sticking probability and the mutual forces of interaction between particles. The van der Waals attraction and Born repulsion forces are included in the calculation of the rate of collisions between two electrically neutral aerosol particles. The overall interaction potential between two particles is calculated through the integration of the inter-molecular potential, modeled as the Lennard-Jones 6-12 potential, under the assumption of pairwise additivity. The expression for the overall interaction potential in terms of the Hamaker constant and the molecular diameter can be found in Appendix 1. The motion of a particle can no longer be assumed to be... 

Here /is the ionization potential of the quenching molecule and = [c/(Q) + (/(I )] /2 is the distance of closest approach of the collision pair, where the d values are taken as Lennard-Jones collision diameters deduced from viscosity measurements. Thus a plot of In (cr ) versus In aQfj j HI ) would be predicted to be linear. This model also predicts some variability for different v vibrational levels due to Franck-Condon effects, but this can be ignored in the present experiment where mainly the v = 32 level is excited by the 532-nm source. 

Fig. 12.2. Potential of mean force fV for two different solute-solvent diameters. Rough estimates of the effective elastic collision diameters of iodine in inert solvents can be obtained from the known Lennard-Jones o parameters. The mean force potential is also compared with the Morse potential, K...

Here Op is the quenching cross-section of molecules P by molecule Q, p is the collision-reduced mass, p the quencher polariabilizty. Ip and 7p are the ionization potentials of P and Q, respectively, and Rq the Lennard-Jones collision diameter. C is a constant, and A is given by... 

Following standard practice the realistic potential elastic collision diameters (Sch3CF3 m) have been estimated using Lennard-Jones collision diameters (o ch3CF3-m) and the Lennard-Jones universal... 

The diameters of solvents play a role in theoretical considerations, such as the application of the scaled particle theory. For gaseous solvent molecules, the collision diameter, o, is related to the Lennard-Jones pair potential energy,... 

Where r Is the potential cut-off distance, the prime denoting differentiation, e Is the well depth and o the molecular collision diameter of the Lennard-Jones Interaction, are the polar angles specifying the orientation of molecule 1, and la = - a. The final term In (2) ensures that the... 

All penetrant molecules were represented as single, spherical, united-atom sites whose short-range interactions with the polymer atoms are described through a 9-6 Lennard-Jones (L-J) potential. The values of collision diameters, a, and the well depths, e, have already been reported in the literature for the COMPASS force field. The nonbonded interactions were truncated at 9.5 A. AU polymer atoms were represented explicitly and the potential function and atomic parameters can be found elsewhere. The reported values correspond to the sampling over five structures and the error bars to standard deviations of the distributions of the calculated values. 

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