Semiempirical Thermodynamic Analysis

If a portion of a titration curve represents thermodynamic equilibrium, it may be treated by standard methods of thermodynamic analysis. If, for example, a protein molecule has 100 dissociable hydrogen ions, and if 

The dissociation constants which one obtains in this way are however of limited value. They provide an algebraic representation of the titration curve, but they do not reflect the properties of individual acidic groups because a species designated as PH, is not a single species, but represents a mixture of molecules in each of which the i protons are attached to a different set of basic groups. 

A much more revealing treatment is one which specifically takes into account the fact that many of the titratable groups of a protein molecule are likely to be intrinsically identical, or very nearly so. Furthermore, it takes into account the one interaction between titratable groups which is 

A detailed derivation of Eqs. (3) and (4) has been given elsewhere (Tanford, 1961a). 

In using these equations we should note that neither Ki nor Kill is a true thermodynamic equilibrium constant because the ratio a ,/(l — xi) is a concentration ratio instead of an activity ratio. Ionic strength is however the only factor which should influence its value measurably (see Section VI, D), and Km, should remain essentially constant over the entire titration curve as long as ionic strength remains constant. It should be noted also that the pH which is ordinarily measured in the laboratory is not exactly — log ae+ if Oh+ is to have the meaning assigned to it in Eq. (2). In dilute aqueous solutions, however, the difference must be quite small (Tanford, 1955a). 

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Molecular descriptors must then be computed. Any numerical value that describes the molecule could be used. Many descriptors are obtained from molecular mechanics or semiempirical calculations. Energies, population analysis, and vibrational frequency analysis with its associated thermodynamic quantities are often obtained this way. Ah initio results can be used reliably, but are often avoided due to the large amount of computation necessary. The largest percentage of descriptors are easily determined values, such as molecular weights, topological indexes, moments of inertia, and so on. Table 30.1 lists some of the descriptors that have been found to be useful in previous studies. These are discussed in more detail in the review articles listed in the bibliography. 

In this chapter, an overview of microkinetic modeling was given. Microkinetic modeling aims at understanding how surface structure and adsorbate properties at the molecular level affect thermodynamic and kinetic phenomena at the macroscale. Inputs to microkinetic modeling via first-principles and semiempirical methods were discussed, followed by an explanation of several microkinetic analysis tools. The modeling of the WGS reaction on platinum was presented as an example of using these tools in the assessment of the surface reaction mechanism. 

Other recently published correlative methods for predicting Tg include the group interaction modeling (GIM) approach of Porter (42), neural networks (43-45), genetic function algorithms (46), the CODESSA (acronym for Comprehensive Descriptors for Structural and Statistical Analysis ) method (47), the energy, volume, mass (EVM) approach (48,49), correlation to the results of semiempirical quantum mechanical calculations of the electronic structure of the monomer (50), and a method that combines a thermodynamic equation-of-state based on lattice fluid theory with group contributions (51). 

Resin chemistry is a mixture of thermodynamic and kinetic knowledge, semiempirical theory, and formulation art. Chemists have sought and continue to seek instrumental and other methods to predict performance prior to plant trials. Methods of chemical analysis, rheological characterization, mechanical analysis, pilot plant evaluation, fracture mechanics, and multivariate statistical analysis all are useful tools. Lab studies are generally designed to bracket the conditions which are likely to be encountered in field trials. When trends are identified, theoretical explanations are formulated. 

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