Debye Hiickel

Debye-Hiickel theory The activity coefficient of an electrolyte depends markedly upon concentration. Jn dilute solutions, due to the Coulombic forces of attraction and repulsion, the ions tend to surround themselves with an atmosphere of oppositely charged ions. Debye and Hiickel showed that it was possible to explain the abnormal activity coefficients at least for very dilute solutions of electrolytes. 

Debye-Hiickel equation Debye-length Condenser capacity... 

At this point an interesting simplification can be made if it is assumed that r, as representing the depth in which the ion discrimination occurs, is taken to be just equal to 1/x, the ion atmosphere thickness given by Debye-Hiickel theory (see Section V-2). In the present case of a 1 1 electrolyte, k = (8ire V/1000eitr) / c /, and on making the substitution into Eq. XV-7 and inserting numbers (for the case of water at 20°C), one obtains, for t/ o in millivolts ... 

In the non-linear differential equation Eq. (43), k is related to the inverse Debye-Hiickel length. The method briefly outlined above is implemented, e.g., in the pro-... 

Table 8.3 Constants of the Debye-Hiickel Equation from 0 to 100°C 8.5...

Although it is not possible to measure an individual ionic activity coefficient,, it may be estimated from the following equation of the Debye-Hiickel theory ... 

At moderate ionic strengths a considerable improvement is effected by subtracting a term bl from the Debye-Hiickel expression b is an adjustable parameter which is 0.2 for water at 25°C. Table 8.4 gives the values of the ionic activity coefficients (for Zi from 1 to 6) with d taken to be 4.6A. 

The values were calculated from the modified Debye-Hiickel equation utilizing the modifications proposed by Robinson and by Guggenheim and Bates ... 

At sufficiently low ionic strengths the activity coefficient of each electrolyte in a mixture is given by the Debye-Hiickel limiting law... 

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 ... 

In Fig. 69 we have been considering a pair of solute particles in pure solvent. We shall postpone further discussion of this question until later. In the meantime we shall review the Coulomb forces in very dilute ionic solutions as they are treated in the Debye-Hiickel theory. 

It is important to realise that whilst complete dissociation occurs with strong electrolytes in aqueous solution, this does not mean that the effective concentrations of the ions are identical with their molar concentrations in any solution of the electrolyte if this were the case the variation of the osmotic properties of the solution with dilution could not be accounted for. The variation of colligative, e.g. osmotic, properties with dilution is ascribed to changes in the activity of the ions these are dependent upon the electrical forces between the ions. Expressions for the variations of the activity or of related quantities, applicable to dilute solutions, have also been deduced by the Debye-Hiickel theory. Further consideration of the concept of activity follows in Section 2.5. 

It can be shown on the basis of the Debye-Hiickel theory that for aqueous solutions at room temperature ... 

For more concentrated solutions (/° 5 >0.3) an additional term BI is added to the equation B is an empirical constant. For a more detailed treatment of the Debye-Hiickel theory a textbook of physical chemistry should be consulted.1... 

Calculation of the Thermodynamic Properties of Strong Electrolyte Solutes The Debye-Hiickel Theory... 

The first assumption of the Debye-Hiickel theory is that is spherically symmetric. With the elimination of any angular dependence, the Poisson equation (expressed in spherical-polar coordinates) reduces to... 

Equation (7.25) can be substituted into equation (7.20) to give a second order differential equation in ijj. In theory, the resulting equation can be solved to give ip as a function of r. However, it has an exponential term in -ip, that makes it impossible to solve analytically. In the Debye-Hiickel approximation, the exponential is expanded in a power series to give... 

Table 7.1 Debye-Hiickel parameters for the activity coefficient, volume, enthalpy, and...

It can be seen from Figure 7.8(b) that the curved lines predicted by the extended form of the Debye-Hiickel equation follow the experimental results to higher ionic strengths than do the limiting law expressions for the (1 1) and (2 1) electrolytes. However, for the (2 2) electrolyte, the prediction is still not very good even at the lowest measured molality.0... 

Experience shows that solutions of other electrolytes behave in a manner similar to the examples we have used. The conclusion we reach is that the Debye-Hiickel equation, even in the extended form, can be applied only at very low concentrations, especially for multivalent electrolytes. However, the behavior of the Debye-Hiickel equation as we approach the limit of zero ionic strength appears to give the correct limiting law behavior. As we have said earlier, one of the most useful applications of Debye-Hiickel theory is to... 

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