Onsager phenomenological coefficients

Irreversible processes are driven by generalized forces, X, and are characterized by transport (or Onsager) phenomenological coefficients, L [21,22], where these transport coefficients, Lip are defined by linear relations between the generalized flux densities,./, which are the rates of change with time of state variables, and the corresponding generalized forces X . 

It follows that the relationship between the Onsager phenomenological coefficient and the traditional diffusion coefficient is... 

The entropy production rate is equal to a sum of products of generalized forces by generalized fluxes. The laws of thermodynamics of irreversible processes enable us to express these fluxes as functions of these forces. When we do not stray too far from the state of equilibrium, where the fluxes and forces are null, linear relations appear between these terms. The coefficients of these linear laws are the Onsager phenomenological coefficients they are combinations of the coefficients of diffusion, viscosity, heat conduction, etc. In conductive media, the electrical resistance also appears as an Onsager coefficient. 

In the derivation of (2.313) the original Onsager phenomenological coefficients were redefined as ... 

Spectral, hemispherical absorbtivity of a surface (—) Onsager phenomenological coefficients (kg s /m )... 

Ternary diffusion coefficients in liquids and solids cannot be found from binary values, but only from experiments. When experiments are not available, which is usually the case, one can make estimates by assuming that the Onsager phenomenological coefficients are a diagonal matrix that is. 

Onsager phenomenological coefficient (Section 7.2) solvent permeability (Section 18.3)... 

According to the Onsager assumption, the square matrix of the phenomenological coefficients... 

The model active transport system described by Dr. Thomas is based on an asymmetric arrangement of two enzymes. A model active transport system was also described by Blumenthal et al. several years ago based on a single enzyme immobilized between asymmetric boundaries [Blumenthal, Caplan, and Kedem, Biophys. J., 7, 735 (1967)]. In the latter case the phenomenological coefficients were measured, and it was possible to demonstrate Onsager symmetry and the correlation between the thermodynamic coefficients and the kinetic constants. 

Onsager s phenomenological coefficients Distance from plane of symmetry, ft. 

Equation (4.11) expresses the central Onsager theorem. It states the symmetry of the phenomenological coefficients (the L matrix) in the absence of magnetic fields. The foundation of this theorem is discussed elsewhere [J. H. Kreuzer (1981) S. R. de Groot, P. Mazur (1962)]. 

The forces Fk involve gradients of intensive properties (temperature, electrochemical potential). The Ljk are called phenomenological coefficients and the fundamental theorem of the thermodynamics of irreversible processes, due originally to Onsager (1931a, b), is that when the fluxes and forces are chosen to satisfy the equation... 

The form of the expressions for the rate of entropy production does not uniquely determine the thermodynamic forces or generalized flows. For an open system, for example, we may define the energy flow in various ways. We may also define the diffusion in several alternative ways depending on the choice of reference average velocity. Thus, we may describe the flows and the forces in various ways. If such forces and flows, which are related by the phenomenological coefficients obeying the Onsager relations, are subjected to a linear transformation, then the dissipation function is not affected by that transformation. 

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. 

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 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. 

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

Equations 3.30 and 3.31 allow for the possibility that each of the flux densities can depend on the differences in both chemical potentials, Afxw and Afis. Four phenomenological coefficients are used in these two equations (Table 3-2). However, by the Onsager reciprocity relation, Lws equals Lsw. Thus, three different coefficients (Lww> Lws> and Lss) are needed to describe the relationship of these two flux densities to the two driving forces. Contrast this with Equation 3.7 [Jj = ufj(-dfLj/dx)] in which a flux density depends on but one force accordingly, only two coefficients are then needed to describe Jw and Js. If the solute were a salt dissociable into two ions, we would have three flux equations (foriM i+, and / ) and three forces (Afxw> Afx+> and Afx ) ... 

In other words, the matrix giving the phenomenological coefficients for the system is symmetric. Another important part of this theory is that one can estimate the entropy generated in an irreversible process. Onsager showed that the local rate of entropy production per unit volume is... 

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