Debye-Hiickel-Onsager conductivity

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]. 

This is possible if the equivalent conductivity is proportional to the square root of the concentration Cq, i.e. if the Debye-Hiickel-Onsager law is obeyed. It is known that this square-root law is also obeyed for non-aqueous solvents as a good approximation, as long as the dielectric constant of the solvent is not less than e = 30. Figure 19 shows the equivalent conductivities as a function of Vm for three examples. If one bears in mind that, because of experimental difficulties, the accuracy of measurements in aqueous solutions is not attained, then the square root law is obeyed to a good approximation. 

Up to concentration of 2 X 10-3 gram-equivalents per litre there is a satisfactory agreement between the results calculated from the Debye-Hiickel-Onsager equation and the actual values of conductance of uni-univalent electrolytes. The validity of this equation has been verified even for uni-bivalent electrolytes, while for bi-bivalent electrolytes there are greater deviations to be observed. 

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. 

Figure 4.101 shows the variation of the equivalent conductivity versus concentration for a number of alkali sulfocyanates in a methanol solvent. The agreement with the theoretical predictions demonstrates the applicability of the Debye-Hiickel-Onsager equation up to at least 2 x 10" mol dm . ... 

W course, the validity of the calculation depends upon whether the theoretical expression for the equivalent conductivity (e.g., the Debye-Hiickel-Onsager equation) is valid in the given concentration range. 

Other workers have looked at typical 2-2 electrolytes at concentrations as low as 1 X 10 mol dm and found that the data fit the Debye-Hiickel-Onsager slope up to around 1 X 10 mol dm. At higher concentrations the observed conductance curve approaches the limiting slope from below, clear evidence of the expected ion association found in typical 2-2 electrolytes (see below). 

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. 

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... 

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 ... 

Finally, it must be recalled that the transport properties of any material are strongly dependent on the molecular or ionic interactions, and that the dynamics of each entity are narrowly correlated with the neighboring particles. This is the main reason why the theoretical treatment of these processes often shows similarities with models used for thermodynamic properties. The most classical example is the treatment of dilute electrolyte solutions by the Debye-Hiickel equation for thermodynamics and by the Debye-Onsager equation for conductivity. 

Debye, Hiickel and Onsager beginning with these conceptions found an interdependence of the equivalent conductance A of a given solution, the con-... 

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