Debye—Hiickel theory

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. 

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 ... 

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 ... 

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... 

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... 

Make a graph of <7, // , and TS against a, and compare the values. E7.5 Use the Debye-Hiickel theory to calculate the activity at 298.15 K of CaCl2 in the following aqueous solutions ... 

Use the Debye-Hiickel theory to calculate 7 for 0.0050 m HC1 dissolved in each of these solvents and make a graph of ln7 against... 

We have problems when we attempt to repeat this calculation in 0.010/77 KC1 and 0.010m KNO, since 7- is a function of the total concentration of ions, and, as we saw in Chapter 7, at m = 0.010 mol-kg-1, 7 differs significantly from one. Debye-Hiickel theory provides a method for calculating 7 for dilute solutions. 

When the Debye-Hiickel theory applies, equation (9.104) becomes... 

Figure 9.5 Extrapolation of the cell emf data of G. A. Linhart [/. Am. Chem. Soc., 41, 1175-1180 (1919)] to obtain E° for the Ag/AgCl half-cell. The dashed line gives the limiting slope as predicted by Debye-Hiickel theory. LHS = Left-hand side of equation see text.

Debye-Hiickel theory 333-50 in electrochemical cells 481-2, 488 and osmotic coefficient 345-8 parameters 342... 

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

Improvements upon the Debye-Hiickel Theory of Ionic Solutions Andersen, H. C. 11... 

Equation (7.44) is known as the third approximation of the Debye-Hiickel theory. Numerous attempts have been made to interpret it theoretically, hi these attempts, either individual simplifying assumptions that had been made in deriving the equations are dropped or additional factors are included. The inclusion of ionic solvation proved to be the most important point. In concentrated solutions, solvation leads to binding of a significant fraction of the solvent molecules. Hence, certain parameters may change when solvation is taken into account since solvation diminishes the number of free solvent molecules (not bonded to the ions). The influence of these and some other factors was analyzed in 1948 by Robert A. Robinson and Robert H. Stokes. 

This is Eq. (7.33), the main equation of the first version of Debye-Hiickel theory. 

Thus we have found that the screening should be more efficient than in the Debye-Hiickel theory. The Debye length l//c is shorter by the factor 1 — jl due to the hard sphere holes cut in the Coulomb integrals which reduce the repulsion associated with counterion accumulation. A comparison with Monte Carlo simulation results [20] bears out this view of the ion size effect [19]. 

The hole correction of the electrostatic energy is a nonlocal mechanism just like the excluded volume effect in the GvdW theory of simple fluids. We shall now consider the charge density around a hard sphere ion in an electrolyte solution still represented in the symmetrical primitive model. In order to account for this fact in the simplest way we shall assume that the charge density p,(r) around an ion of type i maintains its simple exponential form as obtained in the usual Debye-Hiickel theory, i.e.,... 

So far in our revision of the Debye-Hiickel theory we have focused our attention on the truncation of Coulomb integrals due to hard sphere holes formed around the ions. The corresponding corrections have redefined the inverse Debye length k but not altered the exponential form of the charge density. Now we shall take note of the fact that the exponential form of the charge density cannot be maintained at high /c-values, since this would imply a negative coion density for small separations. Recall that in the linear theory for symmetrical primitive electrolyte models we have... 

In the foregoing derivations we have assumed that the true pH value would be invariant with temperature, which in fact is incorrect (cf., eqn. 2.58 of the Debye-Hiickel theory of the ion activity coefficient). Therefore, this contribution of the solution to the temperature dependence has still to be taken into account. Doing so by differentiating ET with respect to T at a variable pH we obtain in AE/dT the additional term (2.3026RT/F) dpH/dT, which if P (cf., eqn. 2.98) is neglected and when AE/dT = 0 for the whole system yields... 

The Debye-Hiickel theory yields the coefficient yc, but the whole subsequent calculation is accompanied by numerous approximations, valid only at high dilutions, so that in the whole region where the theory is valid it may be assumed that y ym yc. 

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