Onsager phenomenological equations

The expression for the diffusion potential can be obtained in a straightforward though hardly brief manner by using the Onsager phenomenological equations to describe the interaction flows. Consido an electrolytic solution consisting of the ionic species and A " and the solvent. When a transport process involves the ions in the system, there are two ionic fluxes,/ and J. Since, however, the ions are solvated, the solvent also participates in the motion of ions and hence there is also a solvent flux Jq. 

Onsager phenomenological equations, 491 Poisson-Boltzmann equation... 

The significance of conformational fluctuations for catalysis has been particularly emphasized by Careri et al. (1975,1979). They used an approach based on the Onsager phenomenological equations. They assumed that weakly bound water molecules play an important role in inducing fluctuations of the backbone by which the protein can exchange free energy with its environment. These fluctuations propagate to the active site and help... 

Onsager postulates [4, 5] the phenomenological equations for irreversible processes given by... 

What has been done so far is to take experimental laws and express them in the form of phenomenological equations, i.e., Eqs. (6.300) and (6.301). Just as the phenomenological equations describing the equilibrium properties of material systems constitute the subject matter of equilibrium thermodynamics, the above phenomenological equations describing the flow properties fall within the purview of nonequilibrium thermodynamics. In this latter subject, the Onsager reciprocity relation occupies a fundamental place (see Section 4.5.7). 

In the absence of gradients of salt concentration and temperature, flows of water and electric current in bentonite clay are coupled through a set of linear phenomenological equations, derived from the theory of irreversible thermodynamics (Katchalsky and Curran, 1967), making use of Onsager s Reciprocal Relations (Groenevelt, 1971) ... 

Equation (3.247) shows that Onsager s reciprocal relations are satisfied in the phenomenological equations (Wisniewski et al., 1976). 

There is no definite sign for Eq. (3.317). When the generalized flows are expressed by linear phenomenological equations with constant coefficients obeying to the Onsager relations... 

Equations (3.331) and (3.332) indicate that the first derivatives of the potentials represent linear phenomenological equations, while the second derivatives are the Onsager reciprocal relations. 

The above phenomenological equations obey Onsager s reciprocal rules, and hence there would be six instead of nine coefficients to be determined. 

This equation is independent of the type of phenomenological relations between fluxes and forces. In contrast, linear phenomenological equations and the Onsager reciprocal relations yield... 

Linear phenomenological equations obey the Onsager reciprocal relations. For the nonlinear region, from the symmetry of the Jacobian of forces versus flows, we have... 

THE LINEAR PHENOMENOLOGICAL EQUATIONS, AND THE ONSAGER RECIPROCITY CONDITIONS... 

It is essential that the relations that are similar to the phenomenological Onsager reciprocal equations are also valid for many types of chemically reactive systems that are far from thermodynamic equilibrium (see Section 2.3.4). 

The complete set of equations, relating fluxes and forces can be written in a matrix form and is called the set of phenomenological equations. It was shown by Onsager [1] that the matrix of the phenomenological proportionality constants is symmetrical ... 

Onsager s thermodynamic equation of motion rests on the linear flux-force relations of transport theory [38]. Thus, to start, we beiefly review these phenomenological equations. 

In previous chapters phenomenological relaxation equations were used together with the Onsager regression hypothesis to compute time correlation functions. In this section we present a microscopic derivation of generalized relaxation equations (Zwan-zig, 1961 Berne, Mori, 1965 and 1971). These equations can be used to compute time-correlation functions under circumstances where the usual phenomenological equations do not apply. 

Onsager (1931) used a set of equations that expresses in an explicit manner the linear dependence of the thermodynamic flows on the thermodynamic forces. These equations, known as the phenomenological equations, can be expressed as... 

Due to the work of Onsager, the matrices of phenomenological coefficients, (Ljj) and (7 ij), are known to be symmetric (Ly =Ljj and = ji)- Essig and Caplan [20] have used the resistive form of the phenomenological equations to define coupling coefficients between the ith and y th flows by... 

According to Onsager s reciprocity theorem Cl], the matrix of the phenomenological equations (7) is symmetric and we have... 

The principle of minimum entropy is restricted to linear phenomenological equations obeying the Onsager relations. In equilibrium thermodynamics, systems tend to maximize the entropy or minimize the free energy. 

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