Potential Energy of Molecular Interactions

For most purposes it is chosen to specify the potential energy of interaction Ep (r) rather than the force of interaction F (r). Suppose that theparticle moves from position r to position oo under the action of a conservative force F. These two functions are then related as  

The hard sphere model is frequently used considering mono-atomic uncharged molecules. However, this model gives a very crude representation of the actual 

2 Phase Space, Distribution Function, Means and Moments 

A concept of principal importance in kinetic theory is the distribution function. The probabilistic distribution function containing the desired information about the spatial distribution of molecules and their velocity distribution is denoted by /(r, c, t). 

The classical phase space is formally defined in terms of generalized coordinates and momenta following the Hamiltonian mechanics notation, because it is in terms of these variables that Liou-viUe s theorem holds. However, in Cartesian coordinates as used in the present section it is usually stiU true that pi = mci under the particular system conditions specified considering the kinetic theory of dilute gases, hence phase space can therefore be defined in terms of the coordinate and velocity variables in this particular case. Nevertheless, in the general case, for example in the presence of a magnetic field, the relation between pi and c is more complicated and the classical formulation is required [115]. 

---

The development of the thermodynamics of thin films is related to the problem of stability of disperse systems. An important contribution to its solving are the works of the Russian scientists Derjaguin and Landau [1] and the Dutch scientists Verwey and Overbeek [2], known today as the DVLO theory. According to their concept the particular state of the thin liquid films is due to the change in the potential energy of molecular interaction in the film and the deformation of the diffuse electric layers. The thermodynamic characteristic of a state of the liquid in the thin film, as shown in Section 3.1, appears to be the dependence of disjoining pressure on film thickness, the n(/t) isotherm. The thermodynamic properties of... 

Fig. 11.1. Schematic representation of potential energy of molecular interaction, as function of intermolecular distance r.

Of the various forms of molecular energy, ideal gas is unique in lacking potential energy of molecular interaction the molecules do not appreciably attract or repulse one another. Since the... 

Equation (4-32) in Table 4.2, the ideal gas equation, is widely applied to pure gases and gas mixtives. This equation neglects molecular size and potential energy of molecular interactions. When each species in a mixture, as well as the mixture, obeys the ideal gas law, both Dalton s law of additive partial pressures and Amagat s law of additive pure species volumes apply. The mixture equation in terms of molal density p/M is... 

The Van der Waals attractive forces are forces of molecular origin, though at their basis lie electrical interactions. By their nature, these forces are caused by molecular polarization under the influence of fluctuations of charge distribution in the neighboring molecule and vice versa. These forces are also known as London s dispersive forces. The potential energy of molecular interaction (the London attraction energy) is equal to... 

The total potential energy of two spherical particles is equal to the sum of electrostatic potential energy and the potential energy of molecular interaction between the particles ... 

A correct calculation of solvation thermodynamics and solution structure is conceivable only in terms of the methods of statistical physics, in particular, the computer experiment schemes, including, in the first place, the molecular dynamics (MD) and the Monte-Carlo (MC) methods [10]. By means of the MD method Newton s classic equations of motion are solved numerically with the aid of a computer assuming that the potential energy of molecular interaction is known. In this manner, the motion of molecules of the liquid may be observed , the phase trajectories found and then the values of the necessary functions are averaged over time and determined. This method permits both the equilibrium and the kinetical properties of the system to be calculated. 

However, changes in the potential energy of intermolecular interactions are not uniquely separable. There is an ambiguity in defining the heat flow for open systems. We may split u into a diffusive part and a conductive part in several ways and define various numbers of heat flows. In the molecular mechanism of energy transport, the energy... 

We start our discussion of the use of potential energy functions in structural biology by a concise review of the quantum mechanical and the empirical basis of our qualitative understanding and quantitative determination of the potential functions of molecular interactions. Such an introductory discussion is necessary because an intelligent and useful application of potential functions... 

These forces are always present and always attractive between particles of the. same nature. They are the result of fluctuations in the dipolar interactions at the molecular level [2,3,13]. The potential energy of this interaction is a function of the separation distance r between dipoles, and has an r dependence. The sum of the interactions between macroscopic objects (as far as molecular dimensions are concerned) yields an interaction energy that is a function of... 

The potential energy of a molecular system in a force field is the sum of individual components of the potential, such as bond, angle, and van der Waals potentials (equation 8). The energies of the individual bonding components (bonds, angles, and dihedrals) are functions of the deviation of a molecule from a hypothetical compound that has bonded interactions at minimum values. 

The correlation of electron motion in molecular systems is responsible for many important effects, but its theoretical treatment has proved to be very difficult. Thus many quantum valence calculations use wave functions which are adjusted to optimize kinetic energy effects and the potential energy of interaction of nuclei and electrons but which do not adequately allow for electron correlation and hence yield excessive electron repulsion energy. This problem may be subdivided into cases of overlapping and nonoverlapping electron distributions. Both are very important but we shall concern ourselves here with only the nonoverlapping case. 

Molecular mechanics calculations use an empirically devised set of equations for the potential energy of molecules. These include terms for vibrational bond stretching, bond angle bending, and other interactions between atoms in a molecule. All of these are summed up as follows ... 

---