Statistical mechanics molecular behavior, description

Boltzmann s expression for S thereby reduces the description of the molecular microworld to a statistical counting exercise, abandoning the attempt to describe molecular behavior in strict mechanistic terms. This was most fortunate, for it enabled Boltzmann to avoid the untenable assumption that classical mechanics remains valid in the molecular domain. Instead, Boltzmann s theory successfully incorporates certain quantal-like notions of probability and indeterminacy (nearly a half-century before the correct quantum mechanical laws were discovered) that are necessary for proper molecular-level description of macroscopic thermodynamic phenomena. 

Molecules are small and light typical linear dimensions are 10 to ICr m, and typical masses are 10 to kg. Hence the number of molecules in a macroscopic system is enonuous. For example, one mole of matter contains 6.022 x 10 molecules (Avogadro s number). Because of these features— smallness, lightness, and numerical abundance— the proper description of behavior at the molecular level and its extrapolation to a macroscopic scale require the special methods of quairtum mechanics aird statistical mechanics. We pursue neitherof these topics here. Instead, we present material nseful for relating molecular concepts to observed thenrrodynamicproperties. 

The limitations imposed on DDL theory as a molecular model by these four basic assumptions have been discussed frequently and remain the subject of current research.In Secs. 1.4 and 3.4 it is shown that DDL theory provides a useful framework in which to interpret negative adsorption and electrokinetic experiments on soil clay particles. This fact suggests that the several differences between DDL theory and an exact statistical mechanical description of the behavior of ion swarms near soil particle surfaces must compensate one another in some way, at least in certain applications. Evidence supporting this conclusion is considered at the end of the present section, whose principal objective is to trace out the broad implications of Eq. 5.1 as a theory of the interfacial region. The approach taken serves to develop an appreciation of the limitations of DDL theory that emerge from the mathematical structure of the Poisson-Boltzmann equation and from the requirement that its solutions be self-consistent in their physical interpretation. TTie limitations of DDL theory presented in this way lead naturally to the concept of surface complexation. 

The theories described in 1 were all considered in general terms either for very simple models of molecules and intermolecular forces or without introducing more realistic descriptions at the molecular level. In this section we describe for equilibrium behavior results obtained by statistical mechanical evaluations and computer simulation for various models of molecular structure and pair interactions.. For background the nature and usefulness of information... 

As the density of the fluid is increased the free motion of molecules is increasingly dominated in the transport process by the interactions among the molecules and especially groups of them. The mean free path becomes smaller and of the order of several molecular diameters. The details of the interactions between the molecules therefore become less important compared to the fact that so many interactions take place. Thus, when the dense liquid state is attained, it seems that quite simple models of the interaction between molecules are adequate for a description of the behavior of the transport properties (see Chapters S and 10). In the extreme case of a fluid near its critical point the specific intermolecular interaction becomes totally irrelevant, since the transport properties of the fluid are determined by the behavior of clusters and their size rather than anything else (see Chapter 6). Thus, the scientific importance of transport properties under these conditions becomes one of seeking to describe the behavior of the property itself through appropriate statistical mechanical theory rather than as a tool to reveal other fundamental information. 

The book is divided into four parts. The first part focuses on the macroscopic properties of physical systems. It begins with the descriptive study of gases and liquids, and proceeds to the study of thermodynamics, which is a comprehensive macroscopic theory of the behavior of material systems. The second part focuses on dynamics, including gas kinetic theory, transport processes, and chemical reaction kinetics. The third part presents quantum mechanics and spectroscopy. The fourth part presents the relationship between molecular and macroscopic properties of systems through the study of statistical mechanics. This theory is applied to the structure of condensed phases. The book is designed so that the first three parts can be studied in any order, while the fourth part is designed to be a capstone in which the other parts are integrated into a cohesive whole. 

The goal of extending classical thermostatics to irreversible problems with reference to the rates of the physical processes is as old as thermodynamics itself. This task has been attempted at different levels. Description of nonequilibrium systems at the hydrodynamic level provides essentially a macroscopic picture. Thus, these approaches are unable to predict thermophysical constants from the properties of individual particles in fact, these theories must be provided with the transport coefficients in order to be implemented. Microscopic kinetic theories beginning with the Boltzmann equation attempt to explain the observed macroscopic properties in terms of the dynamics of simplified particles (typically hard spheres). For higher densities kinetic theories acquire enormous complexity which largely restricts them to only qualitative and approximate results. For realistic cases one must turn to atomistic computer simulations. This is particularly useful for complicated molecular systems such as polymer melts where there is little hope that simple statistical mechanical theories can provide accurate, quantitative descriptions of their behavior. 

Sometimes classical mechanics offers a sufficient description of the dynamical behavior of molecular sysfems. Such a description, however, does not provide the quantum energy levels that are involved in the postulate of equal probabilities. To apply in a classical mechanical framework, fhe postulate of statistical mechanics requires something analogous to quantum states and their energies. We consider the analogy to show that statistical mechanics can be applied without a quantum mechanical analysis however, the primary focus of this chapter uses quantum knowledge about molecules. 

In summary a thermostatic description of the mechanical behavior of homogeneous isotropic elastomeric continua subjected only to large principal stretches is sought. This information provides a guide for relating phenomenological behavior by the tools of statistical mechanics to a molecular model. 

Through approximations and the use of molecular simulation, however, application of statistical mechanics to real fluids has provided enhanced understanding of their behavior and improved description of it through equations of state. Furthermore, the molecular approach of statistical mechanics, as compared to the macroscopic one of traditional thermodynamics, combined with molecular simulation renders it a very useful tool in new applications biochemical processes prediction of matter behavior at and near surfaces, for example in porous materials or thin films ceramic materials polymers etc. 

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