The Debye-Hiickel Theory

According to the Debye-Hiickel theory, the single-ion activity coefficient y, of ion i in a solution of one or more electrolytes is given by 

Im = the ionic strength of the solution on a molality basis, defined by  

T/jermodynam/csandCiiem/stry, second edition, version 3 2011 by Howard DeVoe. Latest version www.chem.umd.edu/thermobook 

Peter Debye made major contributions to various areas of chemistry and physics. 

He was bom in Maastricht, The Netherlands, where his father was foreman in a machine workshop. 

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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... 

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... 

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

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. 

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]. 

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. 

When one takes into account the effects of interaction between the polar head groups using similar degree of the approximation as in the Debye-Hiickel theory, the following relationship results [38] ... 

The Debye-Hiickel theory suggests that the probability of finding ions of the opposite charge within the ionic atmosphere increases with increasing attractive force. 

Although the theory of solutions has been widely used in formulating problems of defects in solids the problems encountered differ in certain respects. The most obvious point is that defects are restricted to discrete lattice sites, whereas the ions in a solution can occupy any position in the fluid. Sometimes no allowance is made for this fact. For example, it has not been demonstrated that at very low concentrations, in the absence of ion-pair effects, the activity coefficients are identical with those of the Debye-Hiickel theory. It can be plausibly argued51 that at sufficiently low concentrations the effect of discreteness is likely to be negligible, but clearly in developing a theory for any but the lowest concentrations the effect should be investigated. A second point... 

Equations (87)-(89) apply in aqueous solutions of two electrolytes in which the interaction potentials are conformal. For example, the assumptions utilized in the extensions of the Debye-Hiickel theory (e.g. water is considered as a continuous dielectric medium of dielectric constant D, that the cation-anion repulsive potential is that of hard spheres, and that all the... 

The case of activity coefficients in solutions is easily but tediously implemented since well-constrained expressions exist, like those produced by the Debye-Hiickel theory for dilute solutions or the Pitzer expressions for concentrated solutions (brines). The interested reader may refer to Michard (1989) for a recent and still reasonably simple account. However simple to handle, activity coefficients introduce analytically cumbersome expressions incompatible with the size of a textbook. Real gas theory demands even more complicated developments. 

The approach that we will follow is known as the Debye-Hiickel theory. The activity laws discussed in the following are derived from a knowledge of electrostatic considerations, and apply to ions in solution that have an energy distribution that follows the well-known Maxwell-Boltzmann law. Strong electrostatic forces affect the behaviour and the mean positions of all ions in solution. 

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