Debye-Hiickel-Onsager conductivity theory

Surf] plot and A refers to the equivalent conductivity of the surfactant counterion at infinite dilution. Models that are more sophisticated are also available for calculating (a i,) from conductivity data at various (T) and ionic strengths these are based on the mass action micellization thermodynamics and the Debye-Hiickel-Onsager conductivity theory [32]. 

Changes to the Debye-Hiickel-Onsager Theory of Conductance... 

Another approach to the conductance of electrolytes, which is less complex than that of Lee and Wheaton, is due to Blum and his co-workers. This theory goes back to the original Debye-Hiickel-Onsager concepts, for it does not embrace the ideas of Lee and Wheaton about the detailed structure around the ion. Instead, it uses the concept of mean spherical approximation of statistical mechanics. This is the rather portentous phrase used for a simple idea, which was fully described in Section 3.12. It is easy to see that this is an approximation because in reality an ionic collision with another ion will be softer than the brick-wall sort of idea used in an MSA approach. However, using MSA, the resulting mathematical treatment turns out to be relatively simple. The principal equation from the theory of Blumet al. is correspondingly simple and can be quoted. It runs... 

The application of Blum s theory to experiment is unexpectedly impressive it can even represent conductance up to 1 mol dm . Figure 4.96 shows experimental data and both theories—Blum s theory and the Debye-Hiickel-Onsager first approximation. What is so remarkable is that the Blum equations are able to show excellent agreement with experiment without taking into account the solvated state of the ion, as in Lee and Wheaton s model. However, it is noteworthy that Blum stops his comparison with experimental data at 1.0 M. 

Various explanations have been given for deviations from the Debye-Hiickel-Onsager equations. A common type of behavior is for the negative slopes of the A versus /c plots to be greater than predicted by the equation that is, the experimental conductivities are lower than predicted by the theory. This has been explained in terms of ion pairing, a concept which was developed by the Danish physical chemist Niels Bjerrum (1879-1958) in 1926. Although most salts, such as sodium chloride, are present in the solid state and in solution as ions and not as covalent species, there is a tendency for them to come together from time to time to form ion pairs. 

When the ions in solvent are forced to move by an external field two effects start to influence conductance. The ions of opposite charge move in opposite directions and their movement is slowed down by the collision of the ionic atmosphere with the solvent molecules. The symmetry of ion distributions is disturbed. These phenomena are called the electrophoretic effect and relaxation effect, respectively. The decrease in conductance resulting from both effects is the basis of the Debye-Hiickel-Onsager theory of conductance [31]... 

A + BA with that determined experimentally from the slope of the equivalent conductance some values are tabulated in Table 3.5. It is clear that the Debye-Hiickel-Onsager theory accounts satisfactorily for the behavior of A at low concentrations. As the ion concentration increases into regions where Debye-Hiickel theory no longer accurately describes electrolyte activity, there are also severe deviations from (3.30). 

Debye-Hiickel-Onsager theory — Onsager equation) Plotting the equivalent conductivity of solutions of strong electrolytes as a function of the square root of concentration gives straight lines according to the Kohlrausch law... 

As the dependency does not include any specific property of the ion (in particular its chemical identity) but only its charge the explanation of this dependency invokes properties of the ionic cloud around the ion. In a similar approach the Debye-Hiickel-Onsager theory attempts to explain the observed relationship of the conductivity on It takes into account the - electrophoretic effect (interactions between ionic clouds of the oppositely moving ions) and the relaxation effect (the displacement of the central ion with respect to the center of the ionic cloud because of the slightly faster field-induced movement of the central ion, -> Debye-Falkenhagen effect). The obtained equation gives the Kohlrausch constant ... 

The symbol A (or A°) represents the maximum theoretical value that the molar conductivity of an electrolyte will approach when diluted indefinitely with an inert solvent. At the beginning of this century Kohlrausch found that the molar conductivity of salts in very dilute aqueous solutions showed a linear relation with the square root of the concentration. This, Kohlrausch s square root law , was incompatible with the Arrhenius electrolytic dissociation theory (q.v.), but it has since been justified by the Debye-Hiickel-Onsager theory of interionic attraction effects, which have been shown to have a dependence. 

Fig. 2. Electrical conductance of 1 1 electrolyte solutions at 298.15 K (data taken from Lobo, 1990). The solid lines and the dashed lines represent, respectively, the predictions of a cube-root linear law (pseudolattice theory) and a square-root linear law in concentration (Debye-Hiickel-Onsager theory).

Incomplete Dissociation into Free Ions. As is well known, there are many substances which behave as a strong electrolyte when dissolved in one solvent, but as a weak electrolyte when dissolved in another solvent. In any solvent the Debye-IIiickel-Onsager theory predicts how the ions of a solute should behave in an applied electric field, if the solute is completely dissociated into free ions. When we wish to survey the electrical conductivity of those solutes which (in certain solvents) behave as weak electrolytes, we have to ask, in each case, the question posed in Sec. 20 in this solution is it true that, at any moment, every ion responds to the applied electric field in the way predicted by the Debye-Hiickel theory, or does a certain fraction of the solute fail to respond to the field in this way In cases where it is true that, at any moment, a certain fraction of the solute fails to contribute to the conductivity, we have to ask the further question is this failure due to the presence of short-range forces of attraction, or can it be due merely to the presence of strong electrostatic forces ... 

One caimot, however, expect the Debye-Huckel-Onsager theory of the nonequilibrium conduction properties of ionic soiutions to fare better at high concentration than the corresponding Debye-Hiickel theory of the equilibrium properties (e.g.. 

In chapter 3, it was shown that the Debye-Hiickel theory for ion-ion interactions is able to account for solution non-ideality in very dilute systems. The same model forms the basis for understanding the concentration dependence of the conductance observed for strong electrolytes. Thus, Onsager [9] showed in 1927 that the limiting conductance law for 1-1 electrolytes has the form... 

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. 

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. 

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. 

Solid electrolytes are not usually solutions of a conducting solute in a solvent matrix. Liquid electrolyte solutions are often sufficiently dilute (1-10 millimolar) to be described by the textbook theories of Debye-Hiickel or Onsager and oppositely charged ions are sufficiently dispersed for interaction between anions and cations to be minimized. By contrast, molten salts are very concentrated (typically 2-20 molar), ion-ion interactions are pronounced, and alternative theories such as that of Fuoss [105] are required. Polymer electrolytes typically have [repeat unit] [cation] ratios, n, in the range 8 to 30, corresponding to 0.7 to 2.5 molar for PEOn LiC104 [106], and ion clustering is an important feature of their behaviour. To account for both the ion-polymer and ion-cluster interactions, Ratner and Nitzan have developed dynamic percolation theory [107]. 

The work of Debye and Erich Hiickel (1896-1880), published in 1923, led to a theory of ionic solutions that explained a number of anomalies concerning conductivities of electrolytic solutions. In 1926, Lars Onsager (1903-76) added the treatment of Brownian motion toward understanding the transport properties of ions in melts, aqueous, and... 

The interionic attraction theory provides a complete explanation for the concentration dependence of conductances in very dilute solutions. Debye and Hiickel s original treatment was improved by Onsager in 1926 and his equation has been convincingly tested over a wide range of conditions. 

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