Fuoss and Onsager

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 1962 Fuoss and Onsager began a revision of their treatment of the conductance of symmetrical electrolytes. In their first paper they considered the potential of total force in the second, the relaxation field in the third, electrophoresis and in the fourth, the hydrodynamic and osmotic terms in the relaxation field (1,2,3,4). In 1965 Fuoss, Onsager, and Skinner (5) combined the results of the four papers and formulated a general conductance equation ... 

The most elaborate treatment of the dissociation of electrolytes in solutions is the one given by Fuoss and Onsager (9, 10). The so-called F.O. equation, applied to I-I associated electrolytes is... 

The early conductance theories given by Debye and Hiickel in 1926, Onsager in 1927 and Fuoss and Onsager in 1932 used a model which assumed all the postulates of the Debye-Hiickel theory (see Section 10.3). The factors which have to be considered in addition are the effects of the asymmetric ionic atmosphere, i.e. relaxation and electrophoresis, and viscous drag due to the frictional effects of the solvent on the movement of an ion under an applied external field. These effects result in a decreased ionic velocity and decreased ionic molar conductivity and become greater as the concentration increases. 

Another empirical extension suggested that there should be a term such as Dcactuai loge actuai This term will come into prominence in the later theories of Fuoss and Onsager (1957 and later) and other workers (see Sections 12.10 and 12.17). 

In aU of these modifications no account was taken of the need to consider cross terms arising from the effect of relaxation on the electrophoretic effect, and from the effect of electrophoresis on relaxation, but they did hint at the form of the conductance theory put forward later by Fuoss and Onsager. 

Fuoss and Onsager subsequently modified their treatment to account for association (see Section 12.12). 

Fuoss has stressed that his equations can only be legitimately applied for concentration ranges where kci < 0.2. This corresponds to approximately 4 x 10 moldm for 1-1 electrolytes, 1 X 10 moldm for 2-2 electrolytes and 4 x 10 moldm for 3-3 electrolytes. The theory also only applies to symmetrical electrolytes, though later workers were able to obtain expressions for unsymmetrical electrolytes. These are much more complex than that of Fuoss and Onsager. 

Smdies in mixed solvents and non-aqueous solvents are outside the scope of this book. But they indicate conclusively that the primitive model used by Fuoss and Onsager up to 1978 is wrong, and that a new model needs to be developed. 

The first chapter of the book sets the stage for many of the topics dealt with later, and, in particular, is a prelude to the development of the two major theoretical topics described in the book, namely the theory of non-ideality and conductance theory. The conventional giants of these fields are Debye and Hiickel with their theory of non-ideality and Debye, Huckel, Fuoss and Onsager with their various conductance equations. These topics are dealt with in Chapters 10 and 12. In addition, the author has included for both topics a qualitative account of modern work in these fields. There is much exciting work being done at present in these fields, especially in the use of statistical mechanics and computer simulations for the theory of nonideality. Likewise some of the advances in conductance theory are indicated. 

Both treatments make use of the same general equations for transport processes in fluids and of the same model to represent the electrolyte solution. However, they lead to somewhat different results due to the manner in which the problem is approached and because of the different boundary conditions employed to evaluate the constants which appear upon integration of the differential equations. We shall give here an account of the conductance theory based on the mathematical approach used by Pitts and shall point out the differences and agreements between his treatment and that of Fuoss and Onsager. The mathematical technique used by the latter authors has been given in detail by Fuoss... 

The distinction between the case v = I and those where v I is due to the fact that the angular dependence of the first term in the r.h.s. of eqn. 5.2.12 is that of the first spherical harmonic. Equation 5.2.15 is analogous to that of Fuoss and Onsager (eqn. 5.2.13) with the T terms set equal to zero. The terms grad-grad, Uiygrad/,5 which give contribu-... 

The alternative approach to the conductance equation for electrolytic solutions followed by Fuoss and Onsager has many points in common to the one employed by Pitts. Some differences are, however, worth mentioning. 

Fuoss and Onsager start their treatment assuming that... 

The expression for AX/X is obtained by a series of successive approximations yielding the terms of different order which contribute to the relaxation field. The first order term arises from eqn. 5.2.13 putting all the T terms equal to zero. Fuoss and Onsager obtain, in this way, an equation equivalent to 5.2.15. The first order expression for is then replaced in the T terms and a further approximation to the perturbed distributions and ionic potentials is calculated. 

It is noteworthy that the majority of the conductance data have been analysed with the equations of Fuoss and Onsager and of Fuoss and Accascina. However, since it has been recently shown that both equations are incomplete and in some cases fail to fit experimental data, we quote here only the improved Fuoss and Hsia result. 

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... 

The limiting slope in eqn. 5.10.3 depends only on the electrophoretic parameter the relaxation factor a having disappeared. A later treatment by Stokes, generalised by Kay and Dye and based on the D-H-O theory as modified by Fuoss and Onsager, - showed... 

The strategies for including these higher order contributions in the conductance equation have been analyzed in detail in the literature (Fem dez-Prini, 1973). At the end of the 1970s there were several alternative equations to the original treatment by Fuoss and Onsager (1957) to account for the effect of concentration on electrolyte conductances the Pitts (1953) equation (P), the Fuoss-Hsia (Fuoss and Hsia, 1967) equation (FH) later modified by Femandez-Prini (1969) (FHFP) and valid only for dilute, binary, symmetrical electrolytes, and the Lee and Wheaton (1978) equation (LW) valid for unsymmetrical electrolytes. 

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