Mathematical treatments statistical thermodynamics

Transport properties are often given a short treatment or a treatment too theoretical to be very relevant. The notion that molecules move when driven by some type of concentration gradient is a practical and easily grasped approach. The mathematics can be minimized. Perhaps the most important feature of the kinetic theory of gases is the recognition that macroscopic properties such as pressure and temperature can be derived by suitable averages of the properties of individual molecules. This concept is an important precursor to statistical thermodynamics. Moreover, the notion of a distribution function as a general... 

All the following considerations refer to pure thermodynamics (phenomenologic, classic and statistic) and do not take into account kinetic aspects of the equilibrium. These aspects will be described in Section 3. It should be noted that statistical mechanics offer a powerful tool for mathematical treatment of both thermodynamics and kinetics. However, the application of statistical mechanics in kinetics will not be demonstrated in this chapter. In the following, it is assumed that the reactional volume is small enough and that the time devoted to the reaction is larger than the time required for establishment of equilibrium. 

Helfand (25,26,27,28,29) has formulated a statistical thermodynamic model of the microphases similar to that of Meier. This treatment, however, requires no adjustable parameters. Using the so-called mean-field-theory approach, the necessary statistics of the molecules are embodied in the solutions of modified diffusion equations. The constraint at the boundary was achieved by a narrow interface approximation which is accomplished mathematically by applying reflection boundary conditions. 

The statistical thermodynamic treatment of the BET theory has the advantage that it provides a satisfactory basis for further refinement of the theory by, say, allowing for adsorbate-adsorbate interactions or the effects of surface heterogeneity. By making the assumptions outlined above, Steele (1974) has shown that the problems of evaluating the grand partition function for the adsorbed phase could be readily solved. In this manner, he arrived at an isotherm equation, which has the same mathematical form as Equation (4.32). The parameter C is now defined as the ratio of the molecular partition functions for molecules in the first layer and the liquid state. 

One of the newer theoretical treatments, based on the pioneering statistical thermodynamic work of McMillan and Mayer (6) y as mathematically formulated by Friedman W, does appear to hold significant promise as a theory of sufficient generality that it may eventually embody other working theories as demonstrated special cases. This theory, known as the cluster integral expansion theory (j ) or simply as cluster theory (9)y has been developed to the point where applications have been made to calculating... 

Most theoretical procedures for deriving expressions for AG iix start with the construction of a model of the mixture. The model is then analyzed by the techniques of statistical thermodynamics. The nature and sophistication of different models vary depending on the level of the statistical mechanical approach and the seriousness of the mathematical approximations that are invariably introduced into the calculation. The immensely popular Flory-Huggins theory, which was developed in the early 1940s, is based on the pseudolattice model and a rather low-level statistical treatment with many approximations. The theory is remarkably simple, explains correctly (at least qualitatively) a large number of experimental observations, and serves as a starting point for many more sophisticated theories. 

In order to take account of the fact that the solvent is made up of discrete molecules, one must abandon the simple hydrodynamically-based model and treat the solvent as a many-body system. The simplest theoretical approach is to focus on the encounters of a specific pair of molecules. Their interactions may be handled by calculating the radial distribution function, whose variations with time and distance describe the behaviour of a pair of molecules which are initially separated but eventually collide. Such a treatment leads (as has long been known) to the same limiting equations for the rate constant as the hydro-dynamically based treatments, including the term fco through which an activation requirement can be expressed, and the time-dependent term in (Equation (2.13)) [17]. The procedure can be developed, but the mathematics is somewhat complex. Non-equilibrium statistical thermodynamics provides an alternative approach [16]. The kinetic theory of liquids provides another model that readily permits the inclusion of a variety of interactions the mathematics is again fairly complex [37,a]. In the computer age, however, mathematical complexity is no bar to progress. Refinement of the model is considered further below (Section (2.6)). 

One of the major problems encountered in writing a book on any particular physical approach to the study of biological processes is that every one of them is dependent on several others, yet not all of them can be treated in the same detail. Quite apart from any purely mathematical simplifications of the treatment given here, it is clearly not possible to provide enough detail of the molecular statistics of diffusion or of statistical thermodynamics, both topics seminal to rate processes, to satisfy a reader who wishes to find a thorough explanation of all concepts used. Hopefully, the references given to sources for treatments of basic principles will be found to be adequate. 

The mathematical foundation for statistical thermodynamics was set by Ludwig Boltzmann in the late 1800s and has been developed to a useful technology for small molecules. Other aspects of thermodynamics still need use of empirically fitted polynomials, but as computers continue to improve, statistical thermodynamics offers a detailed treatment to connect quantized energy levels of molecules to macroscopic thermal properties. There are extensive treatises on statistical thermodynamics but here we have tried to give just the basic essentials as a way to use some of the quantized energy-level formulas we have obtained from quantum mechanics. 

Statistical mechanics is the mathematical means to calculate the thermodynamic properties of bulk materials from a molecular description of the materials. Much of statistical mechanics is still at the paper-and-pencil stage of theory. Since quantum mechanicians cannot exactly solve the Schrodinger equation yet, statistical mechanicians do not really have even a starting point for a truly rigorous treatment. In spite of this limitation, some very useful results for bulk materials can be obtained. 

Parallel with the phenomenological development, an alternative point of view has developed toward thermodynamics, a statistical-mechanical approach. Its philosophy is more axiomatic and deductive than phenomenological. The kinetic theory of gases naturally led to attempts to derive equations describing the behavior of matter in bulk from the laws of mechanics (first classic, then quanmm) applied to molecular particles. As the number of molecules is so great, a detailed treatment of the mechanical problem presents insurmountable mathematical difficulties, and statistical methods are used to derive average properties of the assembly of molecules and of the system as a whole. 

Quantitative models is a heterogeneous group of models expressed in mathematical language. This includes what can be called hard models of general applicability, e.g. thermodynamic models, quantum mechanical models, absolute rate theory, as well as soft models or local models, usually expressed in terms of analogy and similarity, e.g. linear free energy relationships (LFERs), correlations for spectroscopic structural determination, empirical determined kinetic models, and as we shall see, models obtained by statistical treatment of experimental data from properly designed experiments. 

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