Thermodynamic vector entropies

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.

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. 

Schirmer et al. (7.) indicate that the constants and E j may be derived from physical or statistical thermodynamic considerations but do not advise this procedure since theoretical calculations of molecules occluded in zeolites are, at present, at least only approximate, and it is in practice generally more convenient to determine the constants by matching the theoretical equations to experimental isotherms. We have determined the constants in the model by a method of parameter determination using the measured equilibrium data. Defining the entropy constants and energy constants as vectors... 

Since the mixing is the reverse process of separation, the changes in enthalpy and entropy are usually negative and positive, respectively. Therefore the vector appears in the second quadrant on the thermodynamic compass. 

Extended nonequilibrium thermodynamics is not based on the local equilibrium hypothesis, and uses the conserved variables and nonconserved dissipative fluxes as the independent variables to establish evolution equations for the dissipative fluxes satisfying the second law of thermodynamics. For conservation laws in hydrodynamic systems, the independent variables are the mass density, p, velocity, v, and specific internal energy, u, while the nonconserved variables are the heat flux, shear and bulk viscous pressure, diffusion flux, and electrical flux. For the generalized entropy with the properties of additivity and convex function considered, extended nonequilibrium thermodynamics formulations provide a more complete formulation of transport and rate processes beyond local equilibrium. The formulations can relate microscopic phenomena to a macroscopic thermodynamic interpretation by deriving the generalized transport laws expressed in terms of the generalized frequency and wave-vector-dependent transport coefficients. 

It should be stressed that because the vectors J and B lie in orthogonal subspaces, cf. (4.41), (4.174), the product B.J vanishes and B does not appear in (4.178). Consequently, theories of irreversible thermodynamics which try to find fiuxes and forces from the production of entropy overlook the dependence of reaction rate (flux) on force B, see also (4.179) below. 

As we noted below, the equation (4.489) the expressions (4.493) show that dependence of reaction rate on affinity is not so simple [158, 159] as it is assumed in classical non-equilibrium thermodynamics [1, 3, 4, 130] based on entropy production (by chemical reactions), i.e. as a product of fluxes and driving forces (4.178). Projection B of chemical potential vector p, to the subspace W also plays a role in expression for reaction rates J as (4.493) in our example the affinity A is projection of p into orthogonal reaction subspace V only, cf. (4.174). Cf. detailed discussion and criticism in review [108] and references [159, 160]. 

Now, the macroscopic behavior of all systems, whether in equilibrium or nonequilibrium states, is classically described in terms of the thermodynamic variables pressure P, temperature T, specific volume V (volume per mole), specific internal energy U (energy per mole), specific entropy S (entropy per mole), concentration or chemical potential /r, and velocity v. In nonequilibrium states, these variables change with respect to space and/or time, and the subject matter is called nonequilibrium thermodynamics. When these variables do not change with respect to space or time, their prediction falls vmder the subject matter of equilibrium thermodynamics. As a matter of notation, we would indicate a nonequilibrium variable such as entropy hy (r,t), where r is a vector that locates a particular region in space (locator vector) and t is the time, whereas the equilibrium notation would simply be . 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

We shall study the system in Fig. 4. by means of the rininimitm principle of the global entropy production. Let the system be a so called current-tube, which is a tube where the boundary surface is constructed by those vector-lines of the thermodynamic fluxes, which fit on a closed line (for example circle, polygon). 

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