Thermodynamic vector

It is important to recognize that the small subset of matrix equations introduced in the main text (typically, restricted to real matrix elements) will be found sufficient to exploit the geometrical simplicity that underlies equilibrium thermodynamics. Nevertheless, it is useful to introduce the thermodynamic vector geometry in the broader framework of matrix theory and Dirac notation that is broadly applicable to the advanced thermodynamic topics of Chapters 11-13, as well as to many other areas of modem physical chemistry research. 

In the abstract space Ais associated with a given equilibrium state [Pg.345]

Figure 11.2 Thermodynamic vectors for a simple fluid, represented in a two-dimensional diagram in which lengths and angles are expressed in terms of experimental properties for example, cos 0St — (Cy/Cp)1 2 and cos 6Sy = aP(TW/CPpT) 2. The thermodynamically conjugate temperature and volume vectors T), V) are perpendicular, as are the pressure and entropy vectors P), S). A number of thermodynamic relationships among the experimental quantities can be read off directly from the diagram.

The thermodynamic vector a) for the coexistence coordinate cr is therefore composed from the old intensive vectors T), — P) by the equation... 

Table 11.3 displays a variety of geometrical descriptors for each fluid (extending results presented in Sidebar 11.3 for a monatomic ideal gas). These descriptors include the lengths of various thermodynamic vectors ( T, P, S, V ), 0Sy coupling angle (=0TP cf. Fig. 11.2), Gramian M, and minor eigenvalue e2 of the metric M. Sidebar 11.5 describes numerical evaluation of e2. 

Let us first compare the thermodynamic vector lengths for these fluids. Because these lengths carry units, we discuss each vector type in turn, comparing different liquids on the common Si-based scale of responsiveness for the specified thermodynamic variable. 

In contrast to the unit dependence of the thermodynamic vector lengths and metric eigenvalues, the thermodynamic angles are pure dimensionless numbers. Figure 11.9 exhibits the angle 6Sy that each entropy vector S) makes with respect to the volume (abscissa) and temperature (ordinate) axes. 

Let us now examine the deeper geometrical significance of (12.26). For this purpose, it is convenient to select A, B as self-conjugate variables (corresponding to orthonormal thermodynamic vectors), such as the metric eigenmodes Ex, E2 ... 

Gibbs s research interests evolved from engineering to mechanics, and then to thermodynamics, vector analysis and the electromagnetic theory of light, and statistical mechanics. Since his work in thermodynamics and its applications to physical chemistry is likely most relevant to readers of this volume, that work is the subject of my essay. Section II deals with Gibbs s novel development of thermodynamics and Section HI focuses on his application of that thermodynamic theory to various problems in physical chemistry. Section IV then steps back to consider the overall nature of his project and its relevance to philosophy. 

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