Onsager transport coefficient

Pikal MJ (1971) Theory of the Onsager transport coefficients /y and Rij for electrolyte solutions. JPhys Chem 75 3124-3134... 

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. 

In the intermediate regime, this may be recognized as the Green-Kubo expression for the thermal conductivity [84], which in turn is equivalent to the Onsager expression for the transport coefficients [2]. 

The exact treatment yields expressions which have the same form as the expressions given above only the numerical factors are different. The more detailed theory for the diffusion-convection problem between plane walls was developed by Furry, Jones, and Onsager (F10) and that for the column constructed from two concentric cylinders by Furry and Jones (Fll). Recently more attention has been given to the r61e of the temperature dependence of the transport coefficients in column operation (B9, S15). 

In a linear theory, the kinetic coefficients Ly are independent of the forces. They are, however, functions of the thermodynamic variables. In view of the Onsager relations, not only is the L matrix of the transport coefficients symmetric, but the transformed matrix is symmetric as well if the new fluxes are linearly related to the original ones. This also means that the new Ly (i st j) contain diagonal components of the original set. 

Irreversible thermodynamics thus accomplishes two things. Firstly, the entropy production rate EE t allows the appropriate thermodynamic forces X, to be deduced if we start with well defined fluxes (eg., T-VijifT) for the isobaric transport of species i, or (IZT)- VT for heat flow). Secondly, through the Onsager relations, the number of transport coefficients can be reduced in a system of n fluxes to l/2-( - 1 )-n. Finally, it should be pointed out that reacting solids are (due to the... 

The remaining fluxes and forces are independent and thus the Onsager relations Lki = 4 hold. The number of independent transport coefficients is l/2- -(n — 1), With the help of the above conditions, it is possible to verify the symmetry of both matrices L and L [M. Martin, et at. (1988)]... 

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 . 

As the existence of MChA can be deduced by very general symmetry arguments and the effect does not depend on the presence of a particular polarization, one may wonder if something like MChA can also exist outside optical phenomena, e.g. in electrical conduction or molecular diffusion. Time-reversal symmetry arguments cannot be applied directly to the case of diffusive transport, as diffusion inherently breaks this symmetry. Instead, one has to use the Onsager relation. (For a discussion see, e.g., Refs. 34 and 35.) For any generalized transport coefficient Gy (e.g., the electrical conductivity or molecular diffusion tensor) close to thermodynamic equilibrium, Onsager has shown that one can write... 

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. 

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 ... 

Notice that Eq. (A.l la) is identical in form to Eq. (A.IO) so long as one interprets the A(t) s, the X(f) s, and the L s in Eq. (A.lla) as, respectively, the thermodynamic fluxes, forces, and transport coefficients for the process r Tp. However, while the F s in Eq. (A.IO) are external applied forces, the X s in Eq. (A.lla) are internal thermodynamic driving forces. Thus, in adopting Eq. (A.lla), Onsager has boldly postulated that the linear flux-force relations experimentally established for external forces hold as well for internal forces. 

Onsager relations - An important set of equations in the thermodynamics of irreversible processes. They express the symmetry between the transport coefficients describing reciprocal processes in systems with a linear dependence of flux on driving forces. 

One comment should be made regarding the form of the transport equations. In the literature, two-phase flow has often been modeled using Schlogl s equation [50, 51]. This equation is similar in form to Eq. (5.9), but it is empirical and ignores the Onsager cross coefficients. Equations (5.8) and (5.9) stem from concentrated-solution theory and take into account all the relevant interactions. Furthermore, the equations for the liquid-equilibrated transport mode are almost identical to those for the vapor-equilibrated transport mode making it easier to compare the two with a single set of properties (i.e., it is not necessary to introduce another parameter, the elec-trokinetic permeability). 

Here the first equation is the usual Fourier law, the second relates the viscous pressure tensor to the internal variable W, and the last is the evolution of the internal variable. The matrix of the transport coefficients Ly is positive definite with L q = —Lq due to Onsager-Casimir reciprocal rules. 

The modified Enskog equation can easily be generalized to apply to dense mixtures of hard-sphere gases/ The transport coefficients that result satisfy the Onsager reciprocal relations. Thus the principal difficulty in generalizing the Enskog theory to mixtures has now been removed. 

Extended laws are available for the variation with concentration of the transport coefficients of strong and associated electrolyte solutions at moderate to high concentrations. Like the CM calculations, this work is based on the Fuoss-Onsager transport theory. The use of MSA pair distribution functions leads to analytical expressions. Ion association can be introduced with the help of the chemical method. A simplified version of the equations, by taking average ionic diameters, reduces the complexity of the original formulas without really reducing the accuracy of the description and is therefore recommendable for practical use for up to 1-M solutions. 

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