Nonequilibrium Thermodynamics General Relations

G. N. Bochkov and Y. E. Kuzovlev, Non-linear fluctuation relations and stochastic models in nonequilibrium thermodynamics. 1. Generalized fluctuation-dissipation theorem. Physica A 106, 443-J79 (1981). 

The coupling of two different electrokinetic ratios (Estr/p and V/I) through Equation (65) is an illustration of a very general law of reciprocity due to L. Onsager (Nobel Prize, 1968). The general theory of the Onsager relations, of which Equation (65) is an example, is an important topic in nonequilibrium thermodynamics. 

The thermodynamic approach considers micropores as elements of the structure of the system possessing excess (free) energy, hence, micropore formation processes are described in general terms of nonequilibrium thermodynamics, if no kinetic limitations appear. The applicability of the thermodynamic approach to description of micropore formation is very large, because this one is, in most cases, the result of fast chemical reactions and related heat/mass transfer processes. The thermodynamic description does not contradict to the fractal one because of reasons which are analyzed below in Sec. II. C but the nonequilibrium thermodynamic models are, in most cases, more strict and complete than the fractal ones, and the application of the fractal approach furnishes no additional information. If no polymerization takes place (that is right for most of processes of preparation of active carbons at high temperatures by pyrolysis or oxidation of primary organic materials), traditional methods of nonequilibrium thermodynamics (especially nonequilibrium statistical thermodynamics) are applicable. 

These symmetry relationships do not depend on the specific features of any given model but follow quite generally from the linear phenomenological equations of nonequilibrium thermodynamics. Therefore, any linear model that does not predict these relations is likely to be incorrect. 

Chemical process rate equations involve the quantity related to concentration fluctuations as a kinetic parameter called chemical relaxation. The stochastic theory of chemical kinetics investigates concentration fluctuations (Malyshev, 2005). For diffusion of polymers, flows through porous media, and the description liquid helium, Fick s and Fourier s laws are generally not applicable, since these laws are based on linear flow-force relations. A general formalism with the aim to go beyond the linear flow-force relations is the extended nonequilibrium thermodynamics. Polymer solutions are highly relevant systems for analyses beyond the local equilibrium theory. 

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. 

Far from thermodynamic equiHbrium we find nonfinear interdependence of thermodynamic fluxes and forces. In this case, the Onsager reciprocal relations are generally not satisfied, and the formafism developed in Chapter 2 is not fuUy applicable for analysis of the state of open systems. Analysis of systems that are far from thermodynamic equilibrium is the subject of nonlinear nonequilibrium thermodynamics. 

Individual film mass transfer coefficients may be determined by the following considerations. According to postulates of nonequilibrium thermodynamics [78], the general equation that relates the flux, J, of the solute to its concentration, C, and its derivative, is [79]... 

The second, more general, treatment is based on nonequilibrium thermodynamics. Fluxes and forces are connected by a matrix. The diagonal elements (the main effects) of this matrix are well-known - for example, the diffusion coefficient (which is the connection between a particle flux under a concentration gradient) or the thermal conductivity (which relates the temperature gradient with the heat flux). One of the non-diagonal elements is the Seebeck coefficient (= thermopower, ]), which relates a temperature gradient with a particle flux. Based on this, general equations are obtained that describe the heat and particle flow in a thermal and concentration profile ... 

Thermodynamics is generally a very broad discipline, and to write an introductory book self-consistently we had to select only certain, typical part. Constitutive equations offer very different models of thermomechanical phenomena in many diverse materials for applications. In this book, intended for students of chemistry and chemical engineering and related fields, we choose only a narrow sector from these immense fields. Namely, we discuss fhe (mainly nonequilibrium) thermodynamics of fluids (i.e., gas or liquid for difference see Sect. 4.8) and their reacting mixture with... 

As Einstein noted (see the introduction to Chapter 1), it is remarkable that the two laws of thermodynamics are simple to state but they relate so many different quantities and have a wide range of applicability. Thermodynamics gives us many general relations between state variables which are valid for any system in equilibrium. In this section we shall present a few important general relations. We will apply them to particular systems in later Chapters. As we shall see in Chapters 15-17, some of these relations can also be extended to nonequilibrium systems that are locally in equilibrium. 

Linear phenomenological laws of nonequilibrium thermodynamics lead to a general relation between mobility Tk and the diffusion coefficient Dk- This relation can be obtained as follows. The general expression for the chemical potential in a field with potential x / is given by + xjtxl/, in which is the... 

In general, the diagonal elements of a positive definite matrix must be positive. In addition, a necessary and sufficient condition for a matrix Lg to be positive definite is that its determinant and all the determinants of lower dimension obtained by deleting one or more rows and columns must be positive. Thus, according to the Second Law, the proper coefficients L k should be positive the cross coefficients, (i 7 k), can have either sign. Furthermore, as we shall see in the next section, the elements Ljk also obey the Onsager reciprocal relations Ljk = Lkj. The positivity of entropy production and the Onsager relations form the foundation for linear nonequilibrium thermodynamics. 

These two relations, called the Saxen relations, were obtained originally by kinetic considerations for particular systems, but by virtue of the formalism of nonequilibrium thermodynamics we see their general validity. 

Thus nonequilibrium thermodynamics gives a unified theory of irreversible processes. Onsager reciprocal relations are general, valid for all systems in which linear phenomenological laws apply. 

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