Thermodynamic vector length

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. 

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.

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. 

Here Jta(x) denotes the a-th component of the stationary vector x of the Markov chain with transition matrix Q whose elements depend on the monomer mixture composition in microreactor x according to formula (8). To have the set of Eq. (24) closed it is necessary to determine the dependence of x on X in the thermodynamic equilibrium, i.e. to solve the problem of equilibrium partitioning of monomers between microreactors and their environment. This thermodynamic problem has been solved within the framework of the mean-field Flory approximation [48] for copolymerization of any number of monomers and solvents. The dependencies xa=Fa(X)(a=l,...,m) found there in combination with Eqs. (24) constitute a closed set of dynamic equations whose solution permits the determination of the evolution of the composition of macroradical X(Z) with the growth of its length Z, as well as the corresponding change in the monomer mixture composition in the microreactor. 

As pointed out in the foregoing, there are two specific peculiarities qualitatively distinguishing these systems from the classical ones. These peculiarities are intramolecular chemical inhomogeneity of polymer chains and the dependence of the composition of macromolecules X on their length l. Experimental data for several nonclassical systems indicate that at a fixed monomer mixture composition x° and temperature such dependence of X on l is of universal character for any concentration of initiator and chain transfer agent [63,72,76]. This function X(l), within the context of the theory proposed here, is obtainable from the solution of kinetic equations (Eq. 62), supplemented by thermodynamic equations (Eq. 63). For heavily swollen globules, when vector-function F(X) can be presented in explicit analytical form... 

Singular system. It is found from Eqs. (16) and (17) that only a process with AH = 0 and AS 0 is able to constitute a process system by itself. Then it can proceed spontaneously without tranformation of exergy with other processes, as schematically shown in Fig. 17 (b). Of couse its vector appears vertically on the thermodynamic compass, as shown in Fig. 17 (a) and its length corresponds to the magnitude of the exergy destruction in the system. 

In the following chapters of this book, we will discuss fluorescence measurements performed mainly on PE solutions in thermodynamically poor solvents. Therefore, the behavior of PEs in poor solvents is our main sphere of interest, but we will first mention the classical Kuhn treatment of polyelectrolytes in )9-solvents [53]. The potential energy of a polyelectrolyte chain in a given conformation (described by a set of r, position vectors of segments) can be written within the framework of the mean-field Debye-Hiickel (DH) theory [54] as a sum of three contributions the energy corresponding to (i) the entropic elasticity of harmonic bonds, Ui, with bond lengths I, which connect the monomers in the polymer chain. This contribution depends on the set of all position vectors, r, ... 

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