Onsager’s coefficients

The matrix of Onsager s coefficients L, is generally nondiagonal. Therefore, the flux is determined not only by means of one thermodynamic force, but also by taking into account the other forces. A classical example in this respect is the thermoelectric effect. Thermoelectric effects wiU be discussed in greater detail later, for the example of multicomponent diffusion. Onsager also proved that the... 

However, Prigogine s group also estabhshed a more general statement which does not require Onsager s coefficients to be constant that is, it is correct even in nonhnear thermodynamics [2]. To be precise, we can always present the derivative of entropy production with respect to time, in the form of two summands ... 

It has been proven that the first summand (that part of the rate of entropy production change connected with the change of forces) is nonpositive dxthermodynamic force change always aims at decreasing entropy production. We may say that at constant (and symmetric) Onsager s coefficients, we have... 

Moving downward to the molecular level, a number of lines of research flowed from Onsager s seminal work on the reciprocal relations. The symmetry rule was extended to cases of mixed parity by Casimir [24], and to nonlinear transport by Grabert et al. [25] Onsager, in his second paper [10], expressed the linear transport coefficient as an equilibrium average of the product of the present and future macrostates. Nowadays, this is called a time correlation function, and the expression is called Green-Kubo theory [26-30]. 

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. 

The coefficients LgR are Onsager s hnear-response coefficients. The other and so on, characterize the nonlinear response of the... 

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

Delay et al (Ref 12) detd IR absorption spectra in the range 3 to l9u and from the intensities of the bands concluded that the sym form was more abundant in the azides of Ag, Cu, Hg Na but the reverse was true for the azides of Pb Tl. Gray Wad ding ton (Ref 18) stated th at TlNj crysts are isomor-phous with those of Na Rb azides. The elec conductivity of TIN, is 5.9 x 10 s mho at 275° (Ref 18). Brouty (Ref 10) detd the mean activity coefficient of TIN, by EMF+ measurements and calcd ionic radii of Ti Nj. Conductivity measurements by Brouty (Ref 11) did not agree with Onsager s theory deviations were found at very high dilutions. An electro-chem cell used by Suzuki (Ref 16) gave a Ap1 29S° value of 59.17 kcal/mol for... 

Onsager s principle supplements these postulates and follows from the statistical theory of reversible fluctuations [5]. Onsager s principle states that when the forces and fluxes are chosen so that they are conjugate, the coupling coefficients are... 

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. 

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. 

Onsager s reciprocity relationships are valid. According to Onsager the coefficients are related by... 

L length of separation path column or tube length Lkj coefficient in Onsager s equation of reciprocity L mean external length of molecule or colloid m mass... 

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. 

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... 

Table 7.8 shows the thermal diffusion ratios and thermal diffusion coefficients obtained from Onsager s reciprocal rules for toluene, chlorobenzene, and bromobenzene at 1 atm and at 298 and 308 K. Thermal diffusion or heats of transport may be extremely sensitive to the molecular interactions in solutions (Rowley et al., 1988). 

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

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 ... 

According to the Onsager s relations, three coefficients are to be determined. They are the passive permeability to sodium ZNa, the metabolic reaction coefficient if there is no sodium transport Zr, and the cross-coefficient between the chemical reaction and the sodium flow ZNar. The linear nonequilibrium thermodynamics formulation for the active transport of sodium and the associated oxygen consumption in frog skin and toad urinary bladders are studied experimentally. Sodium flow JNa is taken as positive in the direction from the outer to the inner surface of the tissue. The term JT is the rate of suprabasal oxygen consumption assumed to be independent of the oxygen consumption associated with the metabolic functions. 

Lu and L1 are the straight and cross-coefficients, respectively. By Onsager s reciprocal rules, we have LtJ = L. The electrochemical potential differences between internal i and external e regions are defined by... 

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. 

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