Phenomenological reciprocity

However, these diffusion coefficients are applicable only for diluted binary solutions. In real natural water the processes of molecular diffusion are affected by the temperature, pressure, contents and charge of the other components. This effect is defined by phenomenological reciprocity coefficients in Onsager s linear law (equation 3.9). B.P. Boudreau (2004) believes that in hydrochemistry exist two approaches to the evaluation of such effect from top, i.e., from the position of Onsager s linear law, and bottom, i.e., from the position of Fick s diffusion law. We will limit ourselves to a simpler solution of the problem based on the laws of diffusion and thermodynamics. 

The second addend in parentheses of this equation describes the total effect of all phenomenological reciprocity coefficients, i.e., the second... 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

The kinetic constants of the system enter into the phenomenological L-coefficients, which are parameters of state. According to the reciprocity theorem of Onsager, the cross-coefficients L+r and Lr+ are identical. Now the definition of the efficiency 17 emerges directly from the dissipation function... 

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). 

The upshot is that hypnosis and lucid dreaming both result from oppositely directed changes in the balance of regional activation levels that drive AIM toward a forbidden zone with congruent but reciprocal phenomenological features. 

Although irreversible thermodynamics neatly defines the driving forces behind associated flows, so far it has not told us about the relationship between these two properties. Such relations have been obtained from experiment, and famous empirical laws have been established like those of Fourier for heat conduction, Fick for simple binary material diffusion, and Ohm for electrical conductance. These laws are linear relations between force and associated flow rates that, close to equilibrium, seem to be valid. The heat conductivity, diffusion coefficient, and electrical conductivity, or reciprocal resistance, are well-known proportionality constants and as they have been obtained from experiment, they are called phenomenological coefficients Li /... 

In this equation, just as in Newton s law adapted for friction, the reciprocal of the phenomenological coefficient Ln has been introduced and acts as a friction coefficient, a resistance. Recalling the relations... 

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) ... 

Phenomenology of the crystallization. The conversion versus time curves obtained at three different temperatures are shown in Figure 1. With the synthesis procedure used, the sigmoid curves were characterized by shorter induction periods than the traditional method (11,12). As expected, temperature had a strong effect on the rate of crystallization. The overall crystallization rates may be approximated by the reciprocal of the times of half conversion. From these values an apparent activation energy of 22 1 kcal/mol was obtained. With respect to literature data, this value exceeds that reported, for instance, for zeolite Na-X (1,4) but compares well with the 19.8 kcal/mol found for ZSM-11 (13). 

A local thermodynamic state is determined as elementary volumes at individual points for a nonequilibrium system. These volumes are small such that the substance in them can be treated as homogeneous and contain a sufficient number of molecules for the phenomenological laws to be apphcable. This local state shows microscopic reversibility that is the symmetry of all mechanical equations of motion of individual particles with respect to time. In the case of microscopic reversibility for a chemical system, when there are two alternative paths for a simple reversible reaction, and one of these paths is preferred for the backward reaction, the same path must also be preferred for the forward reaction. Onsager s derivation of the reciprocal rules is based on the assumption of microscopic reversibility. 

Onsager s reciprocal relations state that, provided a proper choice is made for the flows and forces, the matrix of phenomenological coefficients is symmetrical. These relations are proved to be an implication of the property of microscopic reversibility , which is the symmetry of all mechanical equations of motion of individual particles with respect to time t. The Onsager reciprocal relations are the results of the global gauge symmetries of the Lagrangian, which is related to the entropy of the system considered. This means that the results in general are valid for an arbitrary process. 

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

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. 

Thus, the Maxwell-Stefan diffusion coefficients satisfy simple symmetry relations. Onsager s reciprocal relations reduce the number of coefficients to be determined in a phenomenological approach. Satisfying all the inequalities in Eq. (6.12) leads to the dissipation function to be positive definite. For binary mixtures, the Maxwell-Stefan dififusivity has to be positive, but for multicomponent system, negative diffusivities are possible (for example, in electrolyte solutions). From Eq. (6.12), the Maxwell-Stefan diffusivities in an -component system satisfy the following inequality... 

By the Onsager reciprocal relations, the matrix of phenomenological coefficients is symmetric, LXq = LqX. Since the dissipation function is positive, the phenomenological coefficients must satisfy the inequalities... 

We may determine each phenomenological coefficient experimentally. The Onsager reciprocal relations reduce the number of coefficients to be determined. If we substitute Eq. (7.42) into Eq. (7.44), we find that the coefficients Liq and Ly obey the following relations ... 

These equations obey the Onsager reciprocal relations, which state that the phenomenological coefficient matrix is symmetric. The coefficients Lqq and Lu arc associated with the thermal conductivity k and the mutual diffusivity >, respectively. In contrast, the cross coefficients Llq and Lql define the coupling phenomena, namely the thermal diffusion (Soret effect) and the heat flow due to the diffusion of substance / (Dufour effect). 

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

The matrix of the phenomenological coefficients must be positive definite for example, for a two-flow system, we have L0 > 0, Ip >0, and Z/.p Z,pZpo > 0.1,0 shows the influence of substrate availability on oxygen consumption (flow), and Ip is the feedback of the phosphate potential on ATP production (flow). The cross-coupling coefficient Iop shows the phosphate influence on oxygen flow, while Zpo shows the substrate dependency of ATP production. Experiments show that Onsagers s reciprocal relations hold for oxidative phosphorylation, and we have Iop = Zpo. 

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... 

These reciprocity relations are derived in Section 6.5. It should be clearly recognized that Eq. (6.3.6) holds only if the phenomenological relations involve conjugate fluxes and forces. If nonconjugate quantities are used, and Lji are... 

Prove that the phenomenological coefficients L12 and L21 in Eq. (6.8.1) satisfy the Onsager Reciprocity Condition. 

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). 

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