Onsager equation derivation

Equation (2.39) leads to the prediction that AA should be proportional to p2. For a bulk solvent, this can be considered as a molecular equivalent of the well-known Onsager formula derived for the continuum dielectric model [12],... 

Various treatments of these effects have been developed over a period of years. The conductance equations of Fuoss and Onsager l, based on a model of a sphere moving through a continuum, are widely used to interpret conductance data. Similar treatments n 3, as well as more rigorous statistical mechanical approaches 38>, will not be discussed here. For a comparison of these treatments see Ref. 11-38) and 39>. The Fuoss-Onsager equations are derived in Ref.36), and subsequently modified slightly by Fuoss, Onsager and Skinner in Ref. °). The forms in which these equations are commonly expressed are... 

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 local thermodynamic state is determined as elementary volumes at individual points for a nonequilibrium system. These volumes are small such that the substance in them can be treated as homogeneous and contain a sufficient number of molecules for the phenomenological laws to be apphcable. This local state shows microscopic reversibility that is the symmetry of all mechanical equations of motion of individual particles with respect to time. In the case of microscopic reversibility for a chemical system, when there are two alternative paths for a simple reversible reaction, and one of these paths is preferred for the backward reaction, the same path must also be preferred for the forward reaction. Onsager s derivation of the reciprocal rules is based on the assumption of microscopic reversibility. 

Since A for various concentrations can be obtained from conductance data and the Onsager equation, by one of the methods described in Chap. Ill, it is possible to derive the dissociation function k for the corresponding concentrations. The results obtained for acetic acid in ag ueous solution at 25° are given in Table XXXVIII, and the values of... 

The arguments are the same as those discussed In the elementary derivation of the Htickel-Onsager equation for electrophoresis, see sec. 4.3a (Hi). Smolu-chowskl (loc. clt) has given a more rigorous derivation for high Ka, which led to... 

A simple treatment of the derivation of the Debye-Huckel-Onsager equation 1927 for symmetrical electrolytes... 

The 35 years of work which went into producing the second modified 1957 Fuoss-Onsager equation did not represent any alteration or improvement on the model, it merely allowed for a few less approximations to be made to the mathematical framework and derivation. However, these did represent a considerable advance on the physically unrealistic approximations in the earher 1927 and 1932 equations. 

When the Aobsvd approach the limiting slope from above this is due to the approximations made in the derivation of the conductance equation. Empirical corrections (see Section 12.8) have been made which postulate higher order terms to be necessary, viz. terms in c, clogc and < with the coefficients of these terms being determined experimentally. But an explanation of this behaviour had to wait until the Fuoss-Onsager equation of 1957 had been formulated (see Section 12.10). 

The shape of the curves shown in Fig. 23—in which three regions are clearly discernible—closely resembles that of the curves derived in previous works for other solvent-solute systems. The explanation for the drop in molar conductivity is to be sought in the Onsager equation and by ion-pair formation as in the previous works. However, the sharp rise in molar conductivity as the salt concentration is increased above 10N is in need of further explanation. 

Since ionic association is an electrostatic effect for equilibrium properties of electrolyte solutions, it may be included in the Debye-Hiickel type of treatment by explicitly retaining further terms in the expansion of the Poisson-Boltzmann relation eqn. 5.2.8. - A similar calculation was attempted for conductance by Fuoss and Onsager. The mathematical approach and the model employed are similar to those used in their previous calculation, but they keep explicitly the exp (—0 y) term in the new calculation. The equation derived for A is... 

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]. 

Onsager s theory can also be used to detemiine the fomi of the flucUiations for the Boltzmaim equation [15]. Since hydrodynamics can be derived from the Boltzmaim equation as a contracted description, a contraction of the flucUiating Boltzmann equation detemiines fluctuations for hydrodynamics. In general, a contraction of the description creates a new description which is non-Markovian, i.e. has memory. The Markov... 

Related to the previous method, a simulation scheme was recently derived from the Onsager-Machlup action that combines atomistic simulations with a reaction path approach ([Oleander and Elber 1996]). Here, time steps up to 100 times larger than in standard molecular dynamics simulations were used to produce approximate trajectories by the following equations of motion ... 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

The problem of relative permittivity of mixed solvents (denoted by e for simplicity) was theoretically studied by Debye and Onsager [34], Onsager derived the following equation ... 

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