Onsager’s reciprocal relation

These relations are the same as the parity rules obeyed by the second derivative of the second entropy, Eqs. (94) and (95). This effectively is the nonlinear version of Casimir s [24] generalization to the case of mixed parity of Onsager s reciprocal relation [10] for the linear transport coefficients, Eq. (55). The nonlinear result was also asserted by Grabert et al., (Eq. (2.5) of Ref. 25), following the assertion of Onsager s regression hypothesis with a state-dependent transport matrix. 

MSN.38. 1. Prigogine and G. Severne, On the validity of Onsager s reciprocity relations for strongly coupled systems, Phys. Lett. 6, 173-176 (1963). 

Note 1 Assuming Onsager s reciprocal relations for irreversible processes, ai- Oi = oe- as... 

Spera F.J. and Trial A.F. (1993) Verification of Onsager s reciprocal relations in a molten silicate solution. Science 259, 204—206. 

It is true that in his own work De Donder did not pursue the consequences of nonequilibrium very far. We have to wait until the basic discovery of Onsager s reciprocity relations in 1931 and till the work of Eckart, Meixner, and many others in the 1940s and the 1950s to see thermodynamics of nonequilibrium processes take shape and be integrated into common knowledge. 

The diffusion coefficients Du and D22 are the principal or "self diffusion coefficients and the off-diagonal quantities D12 and D21 are mutual diffusion coefficients. Even when Onsager s reciprocal relations (31) are valid for the appropriate flow equations so that D12 = D21, there are still three diffusion coefficients generally required to describe the diffusion process. It is noted that even if dC Jbx = 0, the flow of Component 1 is linked to that of Component 2 through the term — Di2dC2/dx, and is not zero. 

There are different ways of describing the coupled flow problem in streaming potentials, one is by at Onsager s reciprocity relations and the other is by... 

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) ... 

A more rigorous approach to calculating the diffusion coefficients has been adopted by Kikuchi [165], A binary substitution alloy (s = 3) has been considered with the vacancy mechanism of atom migration. He was the first to take account of the temporal correlations and to obtain expressions for the correlation cofactor fc in the non-ideal systems. The derived coefficients satisfy Onsager s reciprocal relations. 

We can describe irreversibility by using the kinetic theory relationships in maximum entropy formalism, and obtain kinetic equations for both dilute and dense fluids. A derivation of the second law, which states that the entropy production must be positive in any irreversible process, appears within the framework of the kinetic theory. This is known as Boltzmann s H-theorem. Both conservation laws and transport coefficient expressions can be obtained via the generalized maximum entropy approach. Thermodynamic and kinetic approaches can be used to determine the values of transport coefficients in mixtures and in the experimental validation of Onsager s reciprocal relations. 

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). 

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... 

It is useful to replace the complex coefficients of Eqs. (10.81)-(10.83) with the practical transport coefficients they may be evaluated experimentally under conditions in which two of the independent variables, Jv, A ns/cs, and I, are set equal to zero. Such a set of coefficients may be identified with six coefficients from the set of Eq. (10.96). Because of the Onsager s reciprocal relations, the remaining three coefficients may be evaluated as follows ... 

Also, Onsager s reciprocal relations suggest that LpCa = LCap = rtCaLp and ZpH = Z,Hp = nHLp. 

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. 

Onsager s reciprocal relations of irreversible thermodynamics [27-30] imply that if temperature gradients give rise to diffusion velocities (thermal diffusion), then concentration gradients must produce a heat flux. This reciprocal cross-transport process, known as the Dufour effect, provides another additive contribution to q. It is conventional to express the concentration gradients in terms of differences in diffusion velocities by using the diffusion equation, after which it is found that the Dufour heat flux is [5]. 

The identity of (4.3.21] and (minus) (4.3.4 is an illustration of Onsager s reciprocal relations or, for that matter, of Saxen s relation. The minus sign means that and E are in the seune direction when is negative. This is the situation of figs. 4.6 and 4.7, where the cations move from the right to the left. Considering (4.3.101, may also be called the electro-osmotic slip velocity,... 

The number of different proportionality constants in the latter set of equations is reduced by Onsager s reciprocity relation, which states that Ljj = L--. Such a symmetry is also present in these (though not in all) MNET equations. 

The diffusion-thermal effect or the Dufour energy flux eff describes the tendency of a temperature gradient under the influence of mass diffusion of chemical species. Onsager s reciprocal relations for the thermod3mamics of irreversible processes imply that if temperature gives rise to diffusion velocities (the thermal-diffusion effect or Soret effect), concentration gradients must produce a heat flux. This reciprocal effect, known as the Dufour effect, provides an additional contribution to the heat flux [89]. 

Next, note that from its definition 0 S/A is a symmetric matrix. Also, according to Onsager s reciprocal relations [1,3,4], R is a symmetric matrix. Thus, from eqs. (A.39) and (A.31) and the symmetry of M it follows that both and P are symmetric matrices. 

Here, /c, is the chemical potential, with the term in brackets representing the dimensionless driving force. The 3)-- are generalized diffusion coefficients for the pair // in the multicomponent mixture, defined such that they are consistent with Onsager s reciprocity relation (Van de Ree 1967). 

The yus represent the six coefficients of viscosity of a nematic liquid crystal. However, the number of independent coefficients reduces to five if we assume Onsager s reciprocal relations. 

From Onsager s reciprocal relations in irreversible processes it follows that [D] is symmetric, i.e.. 

The hydrodynamic equations of the classical nematic ( 3.1) are applicable to the N, phase as well. There are six viscosity coefficients (or Leslie coefficients) which reduce to five if one assumes Onsager s reciprocal relations. A direct estimate of an effective value of the viscosity of from a director relaxation measurement indicates that its magnitude is much higher than the corresponding value for the usual nematic. 

In physics and chemistry a number of laws, relations, and principles are statements that some action (commonly expressible as an energy input) causes a reaction (commonly expressible as an energy output). Examples are Guldberg and Waage s law of mass action for chemical reactions, Onsager s reciprocal relations for transport processes, and Newton s third law of motion where a force acting on an object results in a matching reaction force. 

Electrophoresis and sedimentation potential also offer a test of predictions of thermodynamics of irreversible processes, provided these are supplemented by classical analysis of the data. Few measurements of sedimentation potential have been reported [1] and the theories due to Kruyt [2], Debye and Huckel [3] and Henry [4] are not in complete agreement. The thermodynamics of irreversible processes [5] may be helpful since the theory does not depend on any model. In the present chapter it is intended (i) to test linear phenomenological relations, (ii) to test the Onsager s reciprocal relation and (iii) to examine the validity of conflicting theories of electrophoresis. 

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