Onsager’s approach

The two terms in the constant B of Kohlrausch s law represent the two correction effects in Onsager s approach the electrophoretic effect where the ion moves against the flux in... 

Onsager s approach, by definition, is valid in the vicinity of equilibrium, and deviations of Cj from c, eq are assumed to be small. The symmetry of the Onsager matrix, = Lji, follows from the principle of microreversibility (see the classical monograph by de Groot and Mazur, 1962). 

This is Flory s approach to the problem of lyotropic liquid crystals. The ordering transition of a system of rodUke molecules firom an isotropic state to an ordered nematic state is treated in terms of lattice models. Essentially, it is the evaluation of the partition function (see p. 215). As with Onsager s approach, in this model steric... 

These artifacts can be avoided in a new modification of the lattice theory [21]. The type of orientational entropy of a solution of rods was precisely defined in [21]. The distribution function of the rods with respect to orientations Sin), similar to the function used in Onsager s approach, was introduced for this purpose. Instead of Eq. (1.16), it is then possible to write the following for the orientation part of the free energy ... 

A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

Kelvin showed the interdependence of these phenomena by thermodynamic analysis, assuming that the irreversible processes were independent of the reversible ones. This approach was later proved theoretically sound using Onsager s concepts of irreversible thermodynamics (9). 

The relationship between fluctuation and dissipation is reminiscent of the reciprocal Onsager relations that link affinity to flux. The two relationships become identical under Onsager s regression hypothesis which states that the decay of a spontaneous fluctuation in an equilibrium system is indistinguishable from the approach of an undisturbed non-equilibrium system to equilibrium. The conclusion important for statistics, is that the relaxation of macroscopic non-equilibrium disturbances is governed by the same (linear) laws as the regression of spontaneous microscopic fluctuations of an equilibrium system. In the specific example discussed above, the energy fluctuations of a system in contact with a heat bath at temperature T,... 

In the quantum mechanical continuum model, the solute is embedded in a cavity while the solvent, treated as a continuous medium having the same dielectric constant as the bulk liquid, is incorporated in the solute Hamiltonian as a perturbation. In this reaction field approach, which has its origin in Onsager s work, the bulk medium is polarized by the solute molecules and subsequently back-polarizes the solute, etc. The continuum approach has been criticized for its neglect of the molecular structure of the solvent. Also, the higher-order moments of the charge distribution, which in general are not included in the calculations, may have important effects on the results. Another important limitation of the early implementations of this method was the lack of a realistic representation of the cavity form and size in relation to the shape of the solute. 

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. 

The key differences between the PCM and the Onsager s model are that the PCM makes use of molecular-shaped cavities (instead of spherical cavities) and that in the PCM the solvent-solute interaction is not simply reduced to the dipole term. In addition, the PCM is a quantum mechanical approach, i.e. the solute is described by means of its electronic wavefunction. Similarly to classical approaches, the basis of the PCM approach to the local field relies on the assumption that the effective field experienced by the molecule in the cavity can be seen as the sum of a reaction field term and a cavity field term. The reaction field is connected to the response (polarization) of the dielectric to the solute charge distribution, whereas the cavity field depends on the polarization of the dielectric induced by the applied field once the cavity has been created. In the PCM, cavity field effects are accounted for by introducing the concept of effective molecular response properties, which directly describe the response of the molecular solutes to the Maxwell field in the liquid, both static E and dynamic E, [8,47,48] (see also the contribution by Cammi and Mennucci). 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

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. 

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... 

As emphasized, the Born—Kirkwood—Onsager (BKO) approach includes only the solute s monopole and dipole interaction with the continuum. That is, the full classical multipolar expansion of the total solute charge distribution is truncated at the dipole term. This simplification of the electronic distribution fails most visibly for neutral molecules whose dipole moments vanish as a result of symmetry. A distributed monopole or distributed dipole model is more... 

As seen in the earlier discussion of Onsager s theory (section 6.2), in the classical approach diffusion is a response to a concentration gradient. If species i is diffusing then the flux of i is given by... 

The foundations for the macroscopic approach to non-equilibrium thermod5mamics are found in Einstein s theory of Brownian movement [1] of 1905 and in the equation of Langevin [2] of 1908. Uhlenbeck and Omstein [3] generalized these ideas in 1930 and Onsager [4, 5] presented his theory of irreversible processes in 1931. Onsager s theory [4] was initially deterministic with little mention of fluctuations. His second paper... 

The effect of the solvent is usually modelled either by the use of the Onsager s self consistent reaction field (SCRF) [20] or by the polarizable continuum method (PCM) [21]. With regard to the relative stability of cytosine tautomers in aqueous solution, these methods provided results [14,15] which, in spite of some discrepancies, are in reasonable agreement with experimental data [3]. However, continuum-based methods do not explicitly take into consideration the local solvent-solute interaction which is instead important in the description of the proton transfer mechanism in hydrogen-bonded systems. A reasonable approach to the problem was recently proposed [22,23] in which the molecule of interest and few solvent molecules are treated as a supermolecule acting as solute, while the bulk of the solvent is represented as a polarizable dielectric. 

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. 

It is important to digress briefly on the meaning of Bjerrum s approach, which has often been the subject of controversy. Onsager considered that one may say that Bjerrum applies different approximations to different regions of space in evaluating the partition function of the electrolytic solution. In no case may Bjerrum association be taken to imply necessarily a chemical association in the way protons and acetate ions associate to yield the new chemical species acetic acid. Bjerrum pointed out that for this chemical process intermediate species between the associated and dissociated states do not exist in measurable concentrations but between free and associated ions intermediate forms exist in finite concentrations. We shall illustrate this point below. This approach deals with that part of the interaction which is underestimated by the approximation of Debye and Huckel and assumes that ions in the inner region may be considered to form a neutral pair, not affecting the rest of the solution. 

Recent calculations by Miles and Watts (7) on aqueous solutions of a variety of metal ions seem to indicate that Equation 11 likely overestimates the spatial range of the saturation effect that is, the bulk value es would be approached faster than predicted by that equation. In fact, as pointed out by Block and Walker, their expression of <(r) is as approximate and arbitrary as Onsager s step function but has the advantage of reflecting the experimental fact of dielectric saturation and leads to mathematically tractable equations. 

In both cases, the quantity As,—S2 At/ can be identified to As,—S2 > where v is the frequency of the transition under scrutiny. This approach therefore predicts a linear relationship between Av and original form, since the coefficients Cg, (G)oHeni and (Oins, are not obtained from first principles. If, for instance, the Block-Walker model is used, an approximate linear relationship between Av and the function 0( s) is predicted. 

The Onsager treatment provides an expression for the electric field dependence of kf. Note that kf = k Yi s. The electric-field-induced increase in the conductivity of electrolytes usually starts nonlinear, followed by a range where the relative conductivity change, Ak/k 0), is linearly dependent on the electric field strength and finally approaches a field-independent saturation value. In the linear range Onsager s theory of diluted weak electrolytes yields... 

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