Fuoss-Onsager equation

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... 

The Fuoss-Onsager-Skinner equation satisfactorily describes the electrolytic conductance of lithium bromide in acetone. Values of 198.1 0.9 Q l cm2 eq l and (3.3 0.1) X I03 are established for A0 and KA, respectively, at 25°C furthermore, a value of 2.53 A is obtained for the sum of the ionic radii ( ). When bromosuccinic acid is added to 10 5 N lithium bromide in acetone, there is a decrease in the specific conductance of lithium bromide rather than the increase that is observed at higher concentrations. As the concentration of bromosuccinic acid is increased, the values obtained for A0 and KA decrease, while those for a increase when the bromosuccinic acid and acetone are considered to constitute a mixed solvent. These results do not permit any simple explanation. When bromosuccinic acid and acetone are considered a mixed solvent, the Fuoss-Onsager-Skinner theory does not describe the system. 

This study was undertaken to determine whether or not the electrolytic conductance of the lithium bromide-bromosuccinic acid-acetone system can be described by the Fuoss-Onsager-Skinner equation (FOS equation)—Equation 2—by treating the system as lithium bromide in a mixed solvent, and to establish values for Ao and KA for lithium bromide in anhydrous acetone with the same equation. The equation requires knowledge of the concentration and corresponding equivalent conductance along with the dielectric constant and viscosity of the solvent and the temperature that is,... 

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 ... 

A computer program for the solution of the FOS equation, which is a modification of the method of Fuoss, Onsager, and Skinner (5), was written in Fortran IV and executed on an IBM System 360/50 (Operating System—H Level) computer. The program uses the method of Wentworth (35) for least-squares... 

Sometimes, the conductivity of the solution may decrease due to the formation of electroneutral ion pairs. Under these conditions, the Fuoss-Onsager equation can be used to calculate the molar conductivity (A) of associated electrolytes [57] ... 

Section 12.9 on post 1950 modem conductance theories for symmetrical electrolytes and Section 12.10 on Fuoss-Onsager s 1957 conductance equation for symmetrical electrolytes can be omitted until earlier sections are assimilated. These two sections deal with more up to date work which is able to be formulated in a straightforward analytical equation. The development behind these theories is complex and only a brief overview of the ideas behind these theories is given. Nonetheless the Fuoss-Onsager 1957 equation has been much used to analyse experimental data. How this is carried out in practice is given in Sections 12.10.1 to... 

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. 

The apparent close fit to the Debye-Hiickel-Onsager equations at low concentrations has to be reassessed in the light of the cross-over predicted by the later Fuoss-Onsager 1957 equation (see Section 12.10.2). 

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). 

In the early 1950s several workers suggested modifications to the 1927 Debye-Hiickel-Onsager equation and to the 1932 Fuoss-Onsager equation. All of these modifications allowed for ko. 

Fuoss-Onsager 1957 Conductance equation for symmetrical electrolytes... 

Pay particular note the direct effect of considering both relaxation, AXreiaxation, and electrophoresis, AXeiectrophoresis, Contributes a term in i.e. they give the S /c term in the Debye-Hiickel-Onsager equation. When the effects of the cross terms are considered they have no term in and thus do not contribute to the Sy/c term. But all five corrections contribute to the other terms in the final Fuoss-Onsager equation. 

The formulation of these modifications to the external field are complex and lengthy and lie in the field of highly advanced mathematics, but eventually they are resolved into the Fuoss-Onsager equation for unassociated symmetrical electrolytes ... 

Also take note the base for the logarithmic term in Equation (12.52) is not specified. This is because the Fuoss-Onsager equation (Equation 12.52) can be quoted in terms of logio or loge. This will affect the expression for Ei, E2, J, J2, and their values (see Appendix 2, Table 12.3). 

The Fuoss-Onsager equation is also often used in the more approximate form ... 

It can be seen that this treatment gives a theoretical justification for the empirical corrections given in Section 12.8 to the 1927 Debye-Hiickel-Onsager and the 1932 Fuoss-Onsager equations. The denominator (1 + >/Cactuai) 1932 Fuoss-Onsager equation is now... 

For a given solvent at a fixed temperature and making the approximations for , and rj given above, the equation reduces to one in terms of the concentration. A and a and so is an equation in two unknowns. However, it is not possible to solve the equation as a simultaneous equation in the two unknowns using values of A at two concentrations. This is because of the complexity of the functional form of the Fuoss-Onsager equation. 

Modern studies use computer fitting of experimental data to the 1957 Fuoss-Onsager equation, from which values of A°, S and Ei A — 2E2 and J can be found. 

Within the range of concentrations for which the Fuoss-Onsager equation is expected to be valid, this equation accounts well for the effects of non-ideality in solutions of symmetrical electrolytes in which there is no ion association. It can thus be taken as a base-line for non-associated electrolytes and any deviations from this predicted behaviour can be taken as evidence of ion association (see Section 12.12). 

The 1957 Fuoss-Onsager equation can be adapted to take account of association of ions to form ion pairs and to account for incomplete dissociation of weak electrolytes. Chemically these are two different types of situation, but physically they are the same, viz. some of the ions are removed from solution by formation of ion pairs, or by formation of undissociated molecular species. The physical manifestation is that not all of the solute will be able to conduct the current, and so the observed conductance will be lower than that predicted by the... 

Fuoss-Onsager 1957 equation. Fuoss later modified his treatment to include the possibility that some ion pairs are conducting (see Section 12.17). 

Determination of A°, Afassodation and a using the Fuoss-Onsager equation for associated electrolytes... 

MANIPULAnON OF THE 1957 FUOSS-ONSAGER EQUATION, AND LATER MODIFICAHONS 505... 

However, this must be seen in the context of the considerable impetus and stimulus which the Fuoss-Onsager treatment of conductance has given to the experimentalist who has striven to find more and more precise methods with which to test the various theories outlined. This has resulted in very considerable improvements being made to conductance apparatus. It has also placed a very detailed emphasis on obtaining precision and accuracy of the measurements themselves. This has been of considerable import when making measurements at very low concentrations where the experimental difficulties are greatest, but where it is important to test the theory in regions where it is expected to be valid. Such expectations have been vindicated by precision low concentration work where confidence can be placed in the accuracy of the conductance equation. This is reminiscent of the impetus to experimentalists after the Debye-Hiickel equation had been put forward. 

Nonetheless, the Fuoss-Onsager 1957 equations for unassociated and associated electrolytes cope with non-ideality extremely well, provided the concentration range is limited to up to ... 

Manipulation of the 1957 Fuoss-Onsager equation, and Later modifications by Fuoss and other workers... 

In the period between 1957 and 1978 various modifications and extensions were made to the Fuoss-Onsager equations for unassociated and associated electrolytes, but there were no major changes in the model. All that these studies had done was to produce modified conductance equations. 

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