Thermodynamic metrics

H-Azepines 1 undergo a temperature-dependent dimerization process. At low temperatures a kinetically controlled, thermally allowed [6 + 4] 7t-cycloaddition takes place to give the un-symmetrical e.w-adducts, e.g. 2.231-248-249 At higher temperatures (100-200°C) the symmetrical, thermodynamically favored [6 + 6] rc-adducts, e.g. 3, are produced. These [6 + 6] adducts probably arise by a radical process, since a concerted [6 + 6] tt-cycloaddition is forbidden on orbital symmetry grounds, as is a thermal [l,3]-sigmatropic C2 —CIO shift of the unsym-metrical [6 + 4] 7t-dimer. 

The thermodynamic transition between different forms as the above described is formally discontinuous. The difference between polymorphs is shown in general also by a different metrical description of the corresponding lattices. 

This says that fj is orthogonal to the usual thermodynamic force X (using the inner product with metric g ). 

McFarland et al. recently [1] published the results of studies carried out on 22 crystalline compounds. Their water solubilities were determined using pSOL [21], an automated instrument employing the pH-metric method described by Avdeef and coworkers [22]. This technique assures that it is the thermodynamic equilibrium solubility that is measured. While only ionizable compounds can be determined by this method, their solubilities are expressed as the molarity of the unionized molecular species, the intrinsic solubility, SQ. This avoids confusion about a compound s overall solubility dependence on pH. Thus, S0, is analogous to P, the octanol/water partition coefficient in both situations, the ionized species are implicitly factored out. In order to use pSOL, one must have knowledge of the various pKas involved therefore, in principle, one can compute the total solubility of a compound over an entire pH range. However, the intrinsic solubility will be our focus here. There was one zwitterionic compound in this dataset. To obtain best results, this compound was formulated as the zwitterion rather than the neutral form in the HYBOT [23] calculations. 

The titration of very weak acids and bases requires the use of strongly acidic or basic solutions. The determination of thermodynamic pKs is considerably more difficult in these media than in water-rich solutions. Thus, problems are always met when attempting to evaluate activity terms. Also, spectrophoto-metric and NMR titrations are frequently subject to perturbations induced by large changes in solvent composition. 

The general proof of this claim will be presented (Chapter 11) after introduction of the metric geometrical formulation of equilibrium thermodynamics, which makes the basis of the claim rather obvious. More general and powerful geometrical methods of... 

While classical phase diagrams provide a powerful methodology for grasping the thermodynamic behavior of few-component systems, it is evident that the restricted 2D or 3D realm of human graphical intuition cannot adequately cope with the complexities of many-component systems. Hence, it is important to find generalized analytical techniques that can accurately represent many-component phase behavior for arbitrary values of c. Such techniques will be considered in the metric geometric representation of multicomponent phenomena (Chapter 12). 

The self-adjoint ( Hermitian ) matrices often play a particularly important role in representing physical phenomena (e.g., as the observables in quantum theory), and they also include the real symmetric matrices to be encountered in the metric geometry of equilibrium thermodynamics. 

The general line-element expression (9.28) allows one to envision possible geometries with fto/i-Euclidean metric [i.e., failing to satisfy one or more of the conditions (9.27a-c)] or with variable metric [i.e., with a matrix M that varies with position in the space, M = M( i )> a Riemannian geometry that is only locally Euclidean cf. Section 13.1]. However, for the present equilibrium thermodynamic purposes (Chapters 9-12) we may consider only the simplest version of (9.28), with metric elements (R R,-) satisfying the Euclidean requirements (9.27a-c). 

Although the linearity of the chain-rule differential expressions (10.5) confers primitive affine-type spatial structure on thermodynamic variables, it does not yet provide a sense of distance or metric on the space (other than what might be displayed in an arbitrarily chosen axis system). In order to bring intrinsic geometrical structure to the thermodynamic space, we need to define the scalar product (R RJ) [(9.29)] that dictates the spatial metric on Ms- The metric on Ms should reflect intrinsic physical properties of the thermodynamic responses, not merely generic chain rule-type mathematical properties of their differential representation. At the same time, we must exhibit how the space Ms is explicitly connected to the physical measurements of thermodynamic responses. Because such measurements assign scalar values to physical properties, it is natural to associate each scalar product of Ms with the scalar value of an experimental measurement. How can this be done ... 

Having made this long detour into vector geometry and metric spaces, the student of thermodynamics will naturally be impatient to learn the missing link that connects these disparate domains, i.e., that associates the scalar products of the geometry domain... 

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