Statistical Theory Treatment

The exact statistical rate equation (67.Ill) involves the transition probabilities k, (8 ), presumed to be known from accur te quantum-mechanical calculations however, approximate data for l nni and, in particular, results from classical and semiclassical 

The equation (79.Ill)is the complete classical (semiclassical) analogue to the quantum-mechanical statistical formulation (67.111) of the rate constant. It represents actually an exact classical (semiclassical) expression, which is equivalent to the corresponding collision theory rate equation (70,111). 

There obviously exists the same formal resemblance between the two classical (semiclassical) formulations (70.Ill) and (79.Ill) as that between the related quantum-mechanical formulations (51.Ill) and (67.III). The statistical expression (79.Ill) may be obtained from the collision theory equation (70.Ill) by a replacement of the excess  

respectively, if k is expressed as a function of either z. III) and (79.Ill), there exists the relation 

In the particular case of a completely separable reaction coordinate, since Z = Z . The classical (semiclassical) 

---

The earlier models (2-5) dealt primarily with the conformation of a single molecule at an interface and apply at very low adsorption densities. More recent treatments (6-10) take into account polymer-polymer and polymer-solvent interactions and have led to the emergence of a fairly consistent picture of the adsorption process. For details of the statistical theories of polymer adsorption, the reader is referred to publications by Lipatov (11), Tadros (12) and Fleer and Scheutjens (13). 

The understanding of isotope effects on chemical equilibria, condensed phase equilibria, isotope separation, rates of reaction, and geochemical and meteorological phenomena, share a common foundation, which is the statistical thermodynamic treatment of isotopic differences on the properties of equilibrating species. For that reason the theory of isotope effects on equilibrium constants will be explored in considerable detail in this chapter. The results will carry over to later chapters which treat kinetic isotope effects, condensed phase phenomena, isotope separation, geochemical and biological fractionation, etc. 

Table 13.1). In the solid P(CH4) > P(CD4) but the curves cross below the melting point and the vapor pressure IE for the liquids is inverse (Pd > Ph). For water and methane Tc > Tc, but for water Pc > Pc and for methane Pc < Pc- As always, the primes designate the lighter isotopomer. At LV coexistence pliq(D20) < Pliq(H20) at all temperatures (remember the p s are molar, not mass, densities). For methane pliq(CD4) < pLiq(CH4) only at high temperature. At lower temperatures Pliq(CH4) < pliq(CD4). The critical density of H20 is greater than D20, but for methane pc(CH4) < pc(CD4). Isotope effects are large in the hydrogen and helium systems and pLIQ/ < pLiQ and P > P across the liquid range. Pc < Pc and pc < pc for both pairs. Vapor pressure and molar volume IE s are discussed in the context of the statistical theory of isotope effects in condensed phases in Chapters 5 and 12, respectively. The CS treatment in this chapter offers an alternative description.

Although SIKIE may well occur in neutral chemistry (e.g., O3 formation), gas phase ion chemistry has shown itself to be a valuable arena for exploring the phenomenon and evaluating emerging theories. For example, one theory of non-mass-dependent KIE indicated that isotopic fractionation cannot ensue directly from symmetry alone. However, such a conclusion would appear to be incorrect, because that is exactly what is happening in the several cases discussed. The error in that analysis arises in the statistical thermodynamic treatment of the reversible association reaction ... 

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]. 

Statistical theories present particularly useful approaches to the quantitative characterization of dynamical phenomena in chemical kinetics. On the one hand, they provide a shortcut to the overall rate of the reaction, avoiding the explicit treatment of the dynamics before and after reaching the reaction bot-... 

In conclusion, it can be stated that a good first estimate of the energy-transfer spectra is obtained from statistical distributions and that to obtain more quantitative theoretical results, certainly the knowledge of the potential surfaces is at least as important as the correct treatment of rotational and vibrational dynamics. Perhaps a suitable result may already be obtained by merely combining the statistical theory with some knowledge of the potential surface crossing regions. 

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... 

Over the last two to three decades randomized concurrently controlled clinical trials have become established as the method which investigators must use to assess new treatments if their claims are to find widespread acceptance. The methodology underpinning these trials is firmly based in statistical theory, and the success of randomized trials perhaps constitutes the greatest achievement of statistics in the second half of the twentieth century (Preface). 

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). 

Essentially, the problem falls into two parts. Firstly, the calculation of the multi-body reaction potential surface and secondly, the determination of the properties of the reaction products from the knowledge of the potential surface. The reverse process of inverting experimental data to yield a potential energy surface is more complicated and has rarely been attempted. The calculation of potential surfaces and of product distributions may be carried out at various levels of sophistication using classical, semi-classical or full quantum mechanical treatments. Gross features of the reaction potential surfaces may be related to various product properties by simplistic model calculations. Statistical theories may also be used in cases where the lifetime of the collision is long enough to justify their use. 

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. 

Statistical mechanical treatments fail to address the dispersion and polarization interactions that are particularly important when hydrophobic reagents such as IPRs are dealt with [46]. The quantitative treatment of these interactions was introduced by Bockris, Bowler-Reed, and Kitchener [47] their work is important for explaining anomalous salting in when the simple electrostatic theory would predict salting onL... 

Although the major interest in experimental and theoretical studies of network formation has been devoted to elastomer networks, the epoxy resins keep apparently first place among typical thermosets. Almost exclusively, the statistical theory based on the tree-like model has been used. The problem of curing was first attacked by Japanese authors (Yamabe and Fukui, Kakurai and Noguchi, Tanaka and Kakiuchi) who used the combinatorial approach of Flory and Stockmayer. Their work has been reviewed in Chapter IV of May s and Tanaka s monograph Their experimental studies included molecular weights and gel points. However, their conclusions were somewhat invalidated by the fact that the assumed reaction schemes were too simplified or even incorrect. It is to be stressed, however, that Yamabe and Fukui were the first who took into account the initiated mechanism of polymerization of epoxy groups (polyetherification). They used, however, the statistical treatment which is incorrect as was shown in Section 3.3. 

H. Yamakawa, Modem Theory of Polymer Solutions, Harper Row, NY (1971). (Thorough statistical thermodynamic treatment of polymer solution properties.)... 

In this section we develop a statistical theory of EELS lineshapes for binary systems. This treatment will be developed around the correlation expansion. 

For situations of overlapping chains, where lateral fluctuations in the segment concentration become rather small, mean-field descriptions become appropriate. The most successful of this type of theoiy is the lattice model of Scheutjens and Fleer (SF-theoiy). In chapter II.5 some aspects of this model were discussed. This theory predicts how the adsorbed amount and the concentration profile 0(z) depend on the interaction parameters and x and on the chain length N. From the statistical-thermodynamic treatment the Helmholtz energy and, hence, the surface pressure ti can also be obtcdned. When n is expressed as a function of the profile 0(z), the result may be written as ... 

A qualitative discussion of the basic macromolecular configurational problem was first given by Meyer (/) and the statistical theory started with dimers on lattices (2). The first theory, given by Fowler and Rushbrooke (2), was then extended by Chang (3) and Miller 4) to molecules occupying three lattice sites — trimers. A treatment for molecules oo upying any number of sites has been developed by Huggins... 

Summarizing, one can say that the lattice theories need improvement and compact macromolecules need more refined treatment. We shall develop in this paper a refined and unified theory of macromolecular solutions with special emphasis on dilute solutions. We shall put our standpoint on the general theory of solutions developed by McMillan and Mayer in 1945 and Kirkwood and Buff in 1951 (9). TTiese theories do not use the lattice model and are more natural for application especially to dilute solutions. The theories extend statistical theories on gases and this is the reason why we used the name gas theories (70) in the beginning of this Introduction. 

---