The statistical mechanical method

While Debye and HUckel recognized the short-range repulsive forces between ions by assuming a hard-core model, the statistical mechanical methods then available did not allow a full treatment of the effects of this hard core. Only the effect on the electrostatic energy was included—not the direct effect of the hard core on thermodynamic properties. 

The methods used to estimate the effects of plasticization on Tg fall into two general classes namely, semi-empirical equations based on considerations of "free volume" and the much more sophisticated theoretical treatments based on statistical thermodynamics. Neither type of approach is able to provide accurate predictions on a consistent basis. The equations based on free volume [30,67,121] have generally been favored over the statistical mechanical methods (see [122] for an example) for use in practical applications, because of their greater simplicity. For example, the Tg of a plasticized polymer can be estimated very roughly by using the following two equations ... 

The statistical-mechanical methods are, however, currently used for a body of related problems, which concern the optical and electrical properties of matter. We mention, e.g., the theories of the dielectric constant by Yvon, Kirkwood, Brown, de Boer et the theory of the index of refraction by Yvon, and... 

As the principle of minimum entropy production has wide range of applicability, it seems useful to derive this principle by the statistical-mechanical method. This approach will show that the minimum entropy production principle holds in a microscopic description of the system. However, since it is rather complicated to deal with general cases, we shall restrict ourselves to the case of a simple system. Following Klein and Meijer we shall consider a system consisting of two identical chambers, I and II. which contain a total number of N molecules of an ideal... 

Following the statistical mechanical method of Grimley and Mott, we have derived an expression evaluating C g(0) for crystals of various impurity content in the absence of adsorbed surface charge ( ) ... 

Lenstra (1977) used data from 33 different proteins to compare four different methods of prediction, those of Argos and co-workers (1976), the stereochemical method of Lim (1974a,b) which is based on the relative positions of hydrophobic and hydrophilic residues, the statistical-mechanical method of Tanaka and Scheraga (1976a,b,c,d), and the statistical method of Nagano... 

A quantitative theory of rate processes has been developed on the assumption that the activated state has a characteristic enthalpy, entropy and free energy the concentration of activated molecules may thus be calculated using statistical mechanical methods. Whilst the theory gives a very plausible treatment of very many rate processes, it suffers from the difficulty of calculating the thermodynamic properties of the transition state. 

This chapter concentrates on describing molecular simulation methods which have a counectiou with the statistical mechanical description of condensed matter, and hence relate to theoretical approaches to understanding phenomena such as phase equilibria, rare events, and quantum mechanical effects. 

To reiterate a point that we made earlier, these problems of accurately calculating the free energy and entropy do not arise for isolated molecules that have a small number of well-characterised minima which can all be enumerated. The partition function for such systems can be obtained by standard statistical mechanical methods involving a summation over the mini mum energy states, taking care to include contributions from internal vibrational motion. 

In Chapter 2, a brief discussion of statistical mechanics was presented. Statistical mechanics provides, in theory, a means for determining physical properties that are associated with not one molecule at one geometry, but rather, a macroscopic sample of the bulk liquid, solid, and so on. This is the net result of the properties of many molecules in many conformations, energy states, and the like. In practice, the difficult part of this process is not the statistical mechanics, but obtaining all the information about possible energy levels, conformations, and so on. Molecular dynamics (MD) and Monte Carlo (MC) simulations are two methods for obtaining this information... 

Statistical mechanics methods such as Cluster Variation Method (CVM) designed for working with lattice statics are based on the assumption that atoms sit on lattice points. We extend the conventional CVM [1] and present a method of taking into account continuous displacement of atoms from their reference lattice points. The basic idea is to treat an atom which is displaced by r from its reference lattice point as a species designated by r. Then the summation over the species in the conventional CVM changes into an integral over r. An example of the 1-D case was done successfully before [2]. The similar treatments have also been done for... 

Various statistical treatments of reaction kinetics provide a physical picture for the underlying molecular basis for Arrhenius temperature dependence. One of the most common approaches is Eyring transition state theory, which postulates a thermal equilibrium between reactants and the transition state. Applying statistical mechanical methods to this equilibrium and to the inherent rate of activated molecules transiting the barrier leads to the Eyring equation (Eq. 10.3), where k is the Boltzmann constant, h is the Planck s constant, and AG is the relative free energy of the transition state [note Eq. (10.3) ignores a transmission factor, which is normally 1, in the preexponential term]. 

Many other approaches for finding a correct structural model are possible. A short description of ab-initio, density functional, and semiempirical methods are included here. This information has been summarized from the paperback book Chemistry with Computation An Introduction to Spartan. The Spartan program is described in the Computer Software section below.65 Another description of computational chemistry including more mathematical treatments of quantum mechanical, molecular mechanical, and statistical mechanical methods is found in the Oxford Chemistry Primers volume Computational Chemistry,52... 

The statistical mechanics of such impurity systems has been treated by Lidiard61-53 and his method has been widely employed in the interpretation of experimental data, e.g. ionic conductivity,6 51 dielectric loss,8 thermoelectric power,16-36 diffusion,31... 

DIMENSIONAL METHODS IN THE STATISTICAL MECHANICS OF IONIC SYSTEMS... 

Dimensional Methods in the Statistical Mechanics of Ionic Systems... 

What was the distinction between quantum chemistry and chemical physics After the Journal of Chemical Physics was established, it was easy to say that chemical physics was anything found in the new journal. This included molecular spectroscopy and molecular structures, the quantum mechanical treatment of electronic structure of molecules and crystals and the problem of chemical binding, the kinetics of chemical reactions from the standpoint of basic physical principles, the thermodynamic properties of substances and calculation by statistical mechanical methods, the structure of crystals, and surface phenomena. 

Stephen J. Paddison received a B.Sc.(Hon.) in Chemical Physics and a Ph.D. (1996) in Physical/Theoretical Chemistry from the University of Calgary, Canada. He was, subsequently, a postdoctoral fellow and staff member in the Materials Science Division at Los Alamos National Laboratory, where he conducted both experimental and theoretical investigations of sulfonic acid polymer electrolyte membranes. This work was continued while he was part of Motorola s Computational Materials Group in Los Alamos. He is currently an Assistant Professor in the Chemistry and Materials Science Departments at the University of Alabama in Huntsville, AL. Research interests continue to be in the development and application of first-principles and statistical mechanical methods in understanding the molecular mechanisms of proton transport in fuel-cell materials. 

Spectroscopic measurements may, in certain cases, yield direct information on these interactions. On the other hand, thermodynamic values, obtained by measuring certain bulk properties of the system, require the aid of statistical mechanical methods to be related to specific interactions between the solute and the solvent. However, the thermodynamic aspects of the solute-solvent interactions reflect the preference of the solute for one solvent over another and, thereby, determine distribution of the solute in a solvent extraction system. 

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