Pair potential energy

The pair potential energy between the guest molecule and any water molecule is related to the force (F) each exerts on the other by, F = — d

molecular center distance between the two. The potential, which itself is a function of separation distance, is unique to every molecular type and is given by... 

Here, the first six terms refer to pair potential energies of interaction and the last four terms represent dipole-dipole or monopole-dipole interaction energies. 

E.B. Smith, The intermolecular pair-potential energy functions of the inert gases, Physica, 73 (1974) 211-225. 

Fig 1. Potential energy curve of the Au2- Circles represent the relativistic density functional theory (RDFT) results, full line represents the fitted empirical pair potential energy function (PEF). 

We have expressed the pair potential energy function (PEF) of the dimer with in the form of the recently developed empirical function, which works well for transition metals [14]. It s form is... 

Corresponding States (CS) The principle of CS applies to conformal fluids [Lehmd, T. L., Jr., and P. S. Chappelear, Ind. Eng. Chem., 60 (1968) 15]. Two fluids are conformal if their intermolecu-lar interactions are equivalent when scaled in dimensionless form. For example, the Lennard-Jones (LJ) intermolecular pair potential energy U can be written in dimensionless form as... 

Adding up the pair-potential energies for the entire cavity we obtain ... 

Over the past 15 years, a large number of papers have appeared dealing with computer simulations of water structure, thanks to the increased capacity and availability of fast computers. Simulations are based on accepting a reasonable expression for the pairwise interaction of water molecules, namely, the pair potential (energy) function. Much has been learned from simulation studies, and the results are valuable hints as to what the structure of liquid water may be, but not necessarily as to what the structure must be. Because of computational limitations, the typical sample of water molecules used in such analyses is about 500. If these molecules formed a small droplet, the radius would be five molecules, half of which would be from the outside layer. Thus, one must expect some dramatic surface effects. Furthermore, as discussed below, the calculations are based on a pairwise potential energy function this is probably a most serious limitation, as also discussed below. 

Debye derived a more general expression from Equation (59) for the interactions between dipolar molecules and induced dipolar molecules (rotating) in 1920. He found that when induction takes place, the pair potential energy between two different dipolar molecules each possessing permanent dipole moments of pi and p2 and polarizabilities a, and ah can be expressed as,... 

Quadrupole induction pair potential energy is less than dipole induction pair potential energy. 

The attraction between hydrocarbons or fluorocarbons in air (mainly van der Waals) increases very much when we place these molecules in water. For example, for two contacting methane molecules in free space the interaction pair potential energy is —2.5 x 10 21J, whereas in water it is —14 x 10 21J. On the other hand, experimental evidence... 

If we can write an equation for the chemical potential of a molecule as a function of pair potentials, p = /(p), and find the derivative of the chemical potential with the number density in terms of pair potential energies, we may solve the above integration. Since we know the p2 = P + kT nX2 expression per molecule from Equation (165), where X2 is usually expressed as the mole fraction or volume fraction, we need to relate the chemical potential of pure gas, p, to the molecular pair potentials and also the mole fraction, X2, to the gas number density, p. As the chemical potential, p2, is the total free energy per mole, it includes the interaction energy, p2, as well as enthalpy (kT) and entropy of mixing (kh X2) contributions. 

The presence of solvent molecules between solute molecules affects all the interaction energies. We have seen that the presence of solvent molecules changes the molecular polarizabilities of solutes in Section 2.5.3 and decreases the strength of van der Waals interactions (Section 2.6.4) and the total intermolecular pair potential energies (Section 2.7.4). When two molecules interact in a condensed liquid medium, there are many solvent molecules interfering in this interaction. Now, this becomes a many-body interaction and we have to consider some important new effects ... 

A primary aim of theoretical work in this field is to obtain the phenomenological transport coefficients in terms of molecular scale processes and in terms of molecular parameters such as the intermolecular pair potential energy. Most of the work prior to 1946 rested on rather crude pictures of the liquid state, and various arbitrary assumptions were made as to the nature of the molecular scale transport processes taking place. Born and Green 3 20 21 and... 

Values for the virial coefficients are derived from experimental measurements which can be conveniently classified as follows low pressnre p-V-T measnrements high pressnre p-V-T measnrements speed of sonnd measurements vaponr pressnre and enthalpy of vaporization measnrements refractive index/dielectric constant measurements and Jonle-Thomson experiments. These will be discussed in Chapter 1.2, and methods of data evalnation described in Chapter 1.5. Much attention has been paid to the correlation of virial coefficient data and the more satisfactory methods are considered in Chapter 1.3, together with a brief discussion of the theoretical calculation of the second virial coefficient from pair potential energy functions which have been derived a priori or by consideration of other dilute gas properties. So far, this calculation is only applicable to molecules with a spherically symmetric intermolecular potential energy function, for which... 

The most satisfactory methods of data correlation are based on sound theory. In the case of the second virial coefficient, this depends on accurate knowledge of the intermolecular pair potential energy function, U(R, coi, oi) which, in general, depends on the orientations, Oh and ofe, as well as the separation, R, of the molecules ... 

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