Conductance equation for symmetrical electrolytes

10 Fuoss-Onsager 1957 Conductance equation for symmetrical electrolytes 

There are very good reasons why there was the long delay of 30 years between formulation of the Debye-Hiickel model, recognition of the necessity to consider the effect of relaxation and 

The physical model is simple, but the mathematics involved is formidable and almost intractable. The problem does not lie in the numerical aspects of the calculation, but in the analytical developments of the types of equations which are required to describe the electrophoretic and relaxation effects and their cross terms. Computing technology will not help here, unless a model were developed to which molecular dynamics could be applied. 

Because of the mathematical intractability of this conductance theory, it is not easy, and perhaps impossible, to demonstrate in a short account the cross-linking between the model and the mathematics. Nor is it possible to give an account of the theory at a level and in the same detail as that given for the Debye-Hiickel theory in Chapter 10. And so only a brief description of the theoretical treatment and the results of conductance theory, post 1950, are given. 

For theories later than the Fuoss-Onsager 1932 treatment it is useful to express the effects of electrophoresis, relaxation and other contributions in a form showing how they modify the external field under which the ions are migrating. 

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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... 

Step 5 The final step in arriving at the conductance equation for symmetrical electrolytes... 

Following the concept underlying the MSA-MAL conductivity equation [3, 32, 33] and by taking into account that the total concentration of electrically conducting particles is molar conductivity in the AMSA for symmetrical electrolytes [13]. The possibility of such modification of the AMSA theory is quite promising for the description and interpretation of thermodynamic and transport properties of electrolyte solutions in a weakly polar solvent. 

Murphy and Cohen have recently derived a conductance equation for un-symmetrical electrolytes with a term linear in concentration (/. chem. Phys., 53, 2173 (1970)). 

The equation for the molar ionic conductivity for a symmetrical electrolyte is ... 

For very dilute solutions, the equivalent conductivity for any electrolyte of concentration c can be approximately calculated using the Debye-Hiickel-Onsager equation, which can be written for a symmetrical (equal charge on cation and anion) electrolyte as... 

Shortly after Debye and Hiickel had presented their momentous work on the free energy of electrolyte solutions, Onsager derived theoretically, the empirical equation proposed by Kohlrausch to represent the molar conductance of an electrolyte solution. For solutions of a single symmetrical electrolyte this equation is given by... 

Equation 5.2.9 differs from that employed by Debye and Huckel in the presence of the quadratic term which they omitted. Conductance theory has dealt so far only with symmetrical electrolytes. For these salts the even powers in the expansion of 5.2.8 do not contribute to the equilibrium properties of their solutions, affecting only their transport properties. 

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. 

To determine the distribution of electric potential, as modified by surface conductance, we again take the electric field to be spherically symmetric and to satisfy the Laplace equation. A thin spherical double layer shell is considered to surround the particle, and the conductivity of this shell is taken to have the mean value cr(. In reality the conductivity in the thin double layer varies continuously. Outside of the double layer shell the bulk conductivity is that of the electrolyte. This electrostatics problem is a straightforward one in which, from the Laplace equation, the solution for the potential is... 

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