Distribution Functions and Thermodynamics

This chapter summarizes the most important relations between thermodynamic quantities and molecular distribution functions for pure liquids. Most of these relations apply to systems obeying the assumption of pairwise additivity for the total potential energy. We shall indicate, however, how to modify the relations when higher-order potentials are to be incorporated in the formal theory. In general, higher-order potentials bring in higher-order MDFs. Since very little is known about the analytical behavior of the latter, such relationships are rarely useful in applications. 

Most of the specific derivations carried out in this section apply to systems of simple spherical particles. We shall also point out the appropriate generalizations for nonspherical particles that do not possess internal rotations. For particles with internal rotations, one needs to take the appropriate average over all conformations. This is discussed in Chapter 8. 

There is a step common to most of the procedures leading to the relations between thermodynamic quantities and the pair distribution function. Therefore, in the next section we derive a general theorem connecting averages of pairwise quantities and the pair distribution function. In fact, we have already quoted one application of this theorem in section 5.5.1. Further applications will appear in subsequent sections of this chapter. 

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In addition to the repulsive part of the potential given by Eq. (4), a short-range attraction between the macroions may also be present. This attraction is due to the van der Waals forces [17,18], and can be modelled in different ways. The OCF model can be solved for the macroion-macroion pair-distribution function and thermodynamic properties using various statistical-mechanical theories. One of the most popular is the mean spherical approximation (MSA) [40], The OCF model can be applied to the analysis of small-angle scattering data, where the results are obtained in terms of the macroion-macroion structure factor [35], The same approach can also be applied to thermodynamic properties Kalyuzhnyi and coworkers [41] analyzed Donnan pressure measurements for various globular proteins using a modification of this model which permits the protein molecules to form dimers (see Sec. 7). 

In summary, the statistical tf-theorem of kinetic theory relates to the Maxwellian velocity distribution function and thermodynamics. Most important, the Boltzmann s //-theorem provides a mechanistic or probabilistic prove for the second law of thermodynamics. In this manner, the //-theorem also relates the thermodynamic entropy quantity to probability concepts. Further details can be found in the standard references [97] [39] [12] [100] [47] [28] [61] [85]. 

In the limit of zero ion size, i.e. as ct 0, the distribution functions and thermodynamic functions in the MS approximation become identical to the Debye-Hiickel limiting law. 

SCF-MI (Self Consistent Field for Molecular Interactions) and non orthogonal Cl were used to determine a water-water interaction potential, from which BSSE is excluded in an a priori fashion. The new potential has been employed in molecular dynamics simulation of liquid water at 25°C. The simulations were performed using MOTECC suite of programs. The results were compared with experimental data for water in the liquid phase, and good accordance was found, both in radial distribution functions and thermodynamic properties, as well as in geometric parameters. 

DISTRIBUTION FUNCTIONS AND THERMODYNAMIC FUNCTIONS OF MULTICOMPONENT SYSTEMS... 

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