Interparticle distance relation

In a solution of a solute in a solvent there can exist noncovalent intermolecular interactions of solvent-solvent, solvent-solute, and solute—solute pairs. The noncovalent attractive forces are of three types, namely, electrostatic, induction, and dispersion forces. We speak of forces, but physical theories make use of intermolecular energies. Let V(r) be the potential energy of interaction of two particles and F(r) be the force of interaction, where r is the interparticle distance of separation. Then these quantities are related by... 

Every gas consists of particles, whether as atoms (such as neon) or as molecules (such as methane). To a relatively good first approximation, any atom can be regarded as a small, incompressible sphere. The reason why we can compress a gas relates to the large separation between the gas particles. The first effect of compressing a gas is to decrease these interparticle distances. 

The following equation relates the interparticle distance (ID) to the volume fraction of the impact modifier (4)) and the weight-average particle size (dlT) [28] ... 

Consider the collision of particles due to wake attraction, as shown in Fig. E3.1. It is assumed that (a) the motion of the leading particle is not affected by the approach of the trailing particle (b) particles are equal-sized, rigid, and spherical and (c) initially, the particles move nearly at their terminal velocities with a very small velocity difference and are separated by a characteristic distance Zo- An empirical relation can be used to describe the effects of the interparticle distance Z and particle Reynolds number Rep on the drag force of the trailing particle as... 

The Van der Waals force that is caused by the molecular vibrations of the material composing micropartides and deposition elements. This is an attraction force that strongly depends on the interparticles distance and on the wavelength (2.) that characterizes the assembly microparticle-deposition element. Relation (4.278) gives a qualitative indication of the value of this force. In this relation, F(u) gives a function which decreases rapidly with the distance (x). -... 

There is no agreement in the literature on the role of the catalyst particle size for methanol oxidation. Two main evidences have been reported one is related to the presence of a maximum in the mass activity as a function of particle size located, for some authors [40], at 3nm and at 2nm for others [41] moreover, for some authors there is no influence of the interparticle distance and particle size on specific activity. If these parameters are not controlling, an increase in mass activity should be observed as the metal surface area is increased [42]. In this context, the metal-... 

In the laboratory frame the motion of the three particles depends on nine variables, three of which define the position of the center-of-mass. Other three coordinates are needed to describe the rotation of the system in the space and therefore the internal motion is described by the three remaining coordinates. For example, in molecular dynamics the potential energy surface in general is calculated and presented using geometrical coordinates, such the interparticle distances, or two bond distances and an angle. But it is convenient and necessary to use different coordinate systems to describe and understand the dynamics of the particles, because of the rotational terms which appear in the full Hamiltonian. In this context, we will present the transformation equations from the interparticle distances to coordinate sets of the hyperspherical and related types, successful in the treatment of the dynamics. 

The interparticle distances are related with Radau-Smith vectors by ... 

Bases of fully exponentially correlated wavefunctions [1, 2] provide more rapid convergence as a function of expansion length than any other type of basis thus far employed for quantum mechanical computations on Coulomb systems consisting of four particles or less. This feature makes it attractive to use such bases to construct ultra-compact expansions which exhibit reasonable accuracy while maintaining a practical capability to visualize the salient features of the wavefunction. For this purpose, exponentially correlated functions have advantages over related expansions of Hylleraas type [3], in which the individual-term explicit correlation is limited to pre-exponential powers of various interparticle distances (genetically denoted r,j). The general features of the exponentially correlated expansions are well illustrated for three-body systems by our work on He and its isoelectronic ions, for... 

---