Statistical thermodynamics of association

The probabilistic model of macromolecular association introduced in the previous section, for the case of large a and n B, may be recast into the formal language in terms of statistical thermodynamics. Recall from Chapter 1 that the chemical potential of a species has two terms, a structural energy (enthalpy) term and a concentration/entropy term  

Here the X can represent A, B, or AB in the current problem. The free energy of the system then is 

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Statistical Thermodynamics of Adsorbates. First, from a thermodynamic or statistical mechanical point of view, the internal energy and entropy of a molecule should be different in the adsorbed state from that in the gaseous state. This is quite apart from the energy of the adsorption bond itself or the entropy associated with confining a molecule to the interfacial region. It is clear, for example, that the adsorbed molecule may lose part or all of its freedom to rotate. 

Occasionally alternative expressions of the PDT (9.5) have been proposed [23-25], These alternatives arise from consideration of statistical thermodynamic manipulations associated with a particular ensemble, and the distinguishing features of those alternative formulae are relics of the particular ensemble considered. On the other hand, relics specific to an ensemble are not evident in the PDT formula (9.5). These alternative formulae should give the same result in the thermodynamic limit. 

Donahue, M. D. Praushitz, J. M. "Statistical Thermodynamics of Solutions in Natural Gas and Petroleum Refining" Research Report 26, Gas Processors Association, Tulsa, OK... 

The problems being addressed in recent work carried out in various laboratories include the fundamental nature of the solute-water intermolecular forces, the aqueous hydration of biological molecules, the effect of solvent on biomolecular conformational equilibria, the effect of biomolecule - water interactions on the dynamics of the waters of hydration, and the effect of desolvation on biomolecular association 17]. The advent of present generation computers have allowed the study of the structure and statistical thermodynamics of the solute in these systems at new levels of rigor. Two methods of computer simulation have been used to achieve this fundamental level of inquiry, the Monte Carlo and the molecular dynamics methods. 

C-C bonds in the actual molecules. The LCT is an analytical molecular-based theory for the statistical thermodynamics of molten polymers, associated with recognizing the degree to which the distinct chemical structures of the individual monomers are relevant. LCT also incorporates free volume and uses the nonrandom... 

Petrucci, S. (1971) Statistical Thermodynamics of Ionic Association and Complexation in Dilute Solutions of Electrolytes. In Ionic Interactions. From Dilute Solutions to Fused Salts S. Petrucci, Ed. Academic Press New York, Vol. 1 pp 117-177. 

Construction of theoretical phase diagrams similar to the diagram in Fig. 2.17 should facilitate the search for major proaches to the analysis of polymer-solvent systems in which equilibria involving isotropic, liquid-crystalline, crystal solvate, and truly crystalline phases are complexly associated. This is valid, since, as indicated in the classic course of statistical thermodynamics of van der Waals and Konstamm [45], physicists and chemists do not need the precise quantitative dependence for the concrete case as much as to establish general types and then to study whether the qualitative differences of these types coincide with the experimentally found types. ... 

By the standard methods of statistical thermodynamics it is possible to derive for certain entropy changes general formulas that cannot be derived from the zeroth, first, and second laws of classical thermodynamics. In particular one can obtain formulae for entropy changes in highly di.sperse systems, for those in very cold systems, and for those associated, with the mixing ofvery similar substances. 

This chapter will describe how we can apply an understanding of thermodynamic behavior to the processes associated with polymers. We will begin with a general description of the field, the laws of thermodynamics, the role of intermolecular forces, and the thermodynamics of polymerization reactions. We will then explore how statistical thermodynamics can be used to describe the molecules that make up polymers. Finally, we will learn the basics of heat transfer phenomena, which will allow us to understand the rate of heat movement during processing. 

In order to understand the thermodynamic issues associated with the nanocomposite formation, Vaia et al. have applied the mean-field statistical lattice model and found that conclusions based on the mean field theory agreed nicely with the experimental results [12,13]. The entropy loss associated with confinement of a polymer melt is not prohibited to nanocomposite formation because an entropy gain associated with the layer separation balances the entropy loss of polymer intercalation, resulting in a net entropy change near to zero. Thus, from the theoretical model, the outcome of nanocomposite formation via polymer melt intercalation depends on energetic factors, which may be determined from the surface energies of the polymer and OMLF. 

The statistical thermodynamic approach of Pitzer (14), involving specific interaction terms on the basis of the kinetic core effect, has provided coefficients which are a function of the ionic strength. The coefficients, as the stoichiometric association constants in our ion-pairing model, are obtained empirically in simple solutions and are then used to predict the activity coefficients in complex solutions. The Pitzer approach uses, however, a first term akin to the Debye-Huckel one to represent nonspecific effects at all concentrations. This weakens somewhat its theoretical foundation. 

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 ... 

Equation (1.44) states that the structural energy increases associated with the creation of defects are offset by entropy increases. The entropy is the number of ways the defects (both interstitials and vacancies) can be arranged within the perfect lattice, and it can be approximated using statistical thermodynamics as... 

Until now, we have focused our attention on those adsorption isotherms that show a saturation limit, an effect usually associated with monolayer coverage. We have seen two ways of arriving at equations that describe such adsorption from the two-dimensional equation of state via the Gibbs equation or from the partition function via statistical thermodynamics. Before we turn our attention to multilayer adsorption, we introduce a third method for the derivation of isotherms, a kinetic approach, since this is the approach adopted in the derivation of the multilayer, BET adsorption isotherm discussed in Section 9.5. We introduce this approach using the Langmuir isotherm as this would be useful in appreciating the common features of (and the differences between) the Langmuir and BET isotherms. 

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