Statistical mechanical treatment

MacKenzie and co-workers [79]. Related is a statistical mechanical treatment by Reiss and co-workers [80] (see also Schonhom [81]). 

As reactants transfonn to products in a chemical reaction, reactant bonds are broken and refomied for the products. Different theoretical models are used to describe this process ranging from time-dependent classical or quantum dynamics [1,2], in which the motions of individual atoms are propagated, to models based on the postidates of statistical mechanics [3], The validity of the latter models depends on whether statistical mechanical treatments represent the actual nature of the atomic motions during the chemical reaction. Such a statistical mechanical description has been widely used in imimolecular kinetics [4] and appears to be an accurate model for many reactions. It is particularly instructive to discuss statistical models for unimolecular reactions, since the model may be fomuilated at the elementary microcanonical level and then averaged to obtain the canonical model. 

Husimi, K., Proc. Phys.-Math. Soc. Japan 22, 264, "Some formal properties of the density matrix." Introduction of the concept of reduced density matrix. Statistical-mechanical treatment of the Hartree-Fock approximation at an arbitrary temperature and an alternative method of obtaining the reduced density matrices are discussed. 

A statistical mechanical treatment shows that the equation of state of a real gas can be expressed as a power series in /Vm as given by equation (A3.3)... 

L. S. Kassel, Statistical mechanical treatment of the activated complex in chemical reactions, J. Chem. Phys. 3, 399 (1935). 

The adsorption of soluble polymers at solid-liquid interfaces is a highly complex phenomenon with vast numbers of possible configurations of the molecules at the surface. Previous analyses of polymer adsorption have ranged in sophistication from very simple applications of "standard" models derived for small molecules, to detailed statistical mechanical treatments of the process. 

In Table 3 are the values of surface tension for the aqueous LAS homolog solutions. Values of molar volume used are those for the pure LAS homolog independent of water. The justification for this comes from the Winsor R model (20, 21) and work by Scriven and Davis (30) who showed that accurate CED values can be obtained from a statistical mechanical treatment of an interface using only 2 or 3 atomic or molecular layers of that interface. For a surfactant solution, the surfactant will predominate in the interface, hence the choice of pure LAS for the solution molar volumes. 

During the last two decades, studies on ion solvation and electrolyte solutions have made remarkable progress by the interplay of experiments and theories. Experimentally, X-ray and neutron diffraction methods and sophisticated EXAFS, IR, Raman, NMR and dielectric relaxation spectroscopies have been used successfully to obtain structural and/or dynamic information about ion-solvent and ion-ion interactions. Theoretically, microscopic or molecular approaches to the study of ion solvation and electrolyte solutions were made by Monte Carlo and molecular dynamics calculations/simulations, as well as by improved statistical mechanics treatments. Some topics that are essential to this book, are included in this chapter. For more details of recent progress, see Ref. [1]. 

For POM, a matrix algorithm for the statistical mechanical treatment of an unperturbed -A-B-A-B- polymer chain with energy correlation between first-neighboring skeletal rotations is described. The results of the unperturbed dimensions are in satisfactory agreement with experimental data. In addition, if the same energy data are used, the results are rather close to those obtained by the RIS scheme usually adopted. The RIS scheme is shown to be also adequate for the calculation of the average intramolecular conformational energy, if the torsional oscillation about skeletal bonds is taken into account in the harmonic approximation. 

N 062 "Statistical Mechanical Treatment of Protein Conformation. I. Conformational Properties of Amino Acids in Proteins ... 

Equation 9-81 is approximate and a more correct statistical mechanical treatment is available.113 114 See also comment on p. 288 about log k (and log /q) being unitless. Employing Eq. 6-14, we may expand Eq. 9-81 as follows ... 

The proper treatment of ionic fluids at low T by appropriate pairing theories is a long-standing concern in standard ionic solution theory which, in the light of theories for ionic criticality, has received considerable new impetus. Pairing theories combine statistical-mechanical theory with a chemical model of ion pair association. The statistical-mechanical treatment is restricted to terms of the Mayer/-functions which are linear in / , while the higher terms are taken care by the mass action law... 

The micellization of surfactants has been described as a single kinetic equilibrium (10) or as a phase separation (11). A general statistical mechanical treatment (12) showed the similarities of the two approaches. Multiple kinetic equilibria (13) or the small system thermodynamics by Hill (14) have been frequently applied in the thermodynamics of micellization (15, 16, 17). Even the experimental determination of the factors governing the aggregation conditions of micellization in water is still a matter of considerable interest (18, 19) and dispute (20). 

The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

How good an approximation is the nonlinear Poisson-Boltzmann equation to more rigorous statistical mechanical treatments ... 

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