Debye-Hiickel, limiting law

For the development of activity coefficient models for electrolyte solutions, the theory of Debye and Huckel is usually the starting point. It can be regarded as an exact equation to describe the behavior of an electrolyte system at infinite dilution. 

For the derivation of the Debye-Huckel limiting law, the following assumptions are made [2]  

1) Only the electrostatic forces between the ions are regarded. All the other forces are negligible. 

2) The electrostatic interaction energies are small in comparison to the thermal energies. 

3) The ions are regarded as punctual charges with a spherical field. 

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

Debye-Hiickel limiting law for 7 applies [equation (7.45)], equation (7.54) can be used to derive a limiting law for 0. We start with... 

Once a value of E° is obtained by extrapolation, the 7 corresponding to each molality can be obtained from equation (9.104). Figure 9.6 is a graph of ln7 calculated from Linhart s results plotted against m1/2. The Debye-Hiickel limiting law value is shown as the dashed line. The agreement is excellent below m1 2 = 0.10 (m = 0.03), which attests to the reliability of Linhart s work and the validity of the Debye-Hiickel limiting law. 

The extrapolation of a graph of LHS against I,1/2 (with the aid of the Debye-Hiickel limiting law) to I 2 = 0 gives an intercept with a value RT/F) n Kw. As a final example, consider the cell... 

Intense ion-ion interactions which are characteristic of salt solutions occur in the concentrated aqueous solutions from which AB cements are prepared. As we have seen, in such solutions the simple Debye-Hiickel limiting law that describes the strength goes up so the repulsive force between the ions becomes increasingly important. This is taken account of in the full Debye-Hiickel equation by the inclusion of a parameter related to ionic size and hence distance of closest approach (Marcus, 1988). 

The Debye-Hiickel limiting law is the least accurate approximation to the actual situation, analogous to the ideal gas law. It is based on the assumption that the ions are material points and that the potential of the ionic atmosphere is distributed from r = 0 to r->oo. Within these limits the last equation is integrated by parts yielding, for constant k, the value ezk/Aite. Potential pk is given by the expression... 

Fig. 1.8 Dependence of the mean activity coefficient y tC of NaCl on the square root of molar concentration c at 25°C. Circles are experimental points. Curve 1 was calculated according to the Debye-Hiickel limiting law (1.3.25), curve 2 according to the approximation aB = 1 (Eq. 1.3.32) curve 3 according to the Debye-Hiickel equation (1.3.31), a = 325nm curve 4 according to the Bates-Guggenheim approximation (1.3.33) curve 5 according to the Bates-Guggenheim approximation + linear term 0.1 C curve 6 according to Eq. (1.3.38) for a = 0.4nm, C = 0.055dm5-mor ...

The Donnan potentials contain the individual ionic activities and cannot be measured by using a purely thermodynamic procedure. In the concentration range where the Debye-Hiickel limiting law is valid, the ionic activities can be replaced by the mean activities. 

If the Debye-Hiickel limiting law is used to evaluate the various activity coefficients in aqueous solution at 25 °C, the last equation becomes... 

The work of Mayer shows that in the limit c —> 0 the expression for log yrs in which all terms in the sum over n > 3 are omitted includes all the terms of order c log c or lower. As in the Mayer solution theory one would hope to build with these terms a low-concentration theory valid over a wider range than the Debye-Hiickel limiting law, which is contained in the cycle terms alone. In the following section we review the detailed evaluation of these... 

The electrostatic function f must contain the Debye-Hiickel limiting law with the parameter... 

In addition, several other forms of correlating equations give comparable fits to the experimental data. One equation uses the higher order limiting law, followed by an empirical polynomial in the square-root of molality. Similarly, another equation uses the Debye-Hiickel limiting law with B set equal to zero, followed by an empirical polynomial in the square-root of molality. Both of these have been discussed in detail elsewhere (Staples and Nuttalf, 1977). 

C. Draw the asymptote predicted by the Debye-Hiickel limiting law for the curve in (a). 

We know from experiment that log 7 is a linear function of I. The value of log 7 + is described well by the Debye-Hiickel limiting law in very dilute solution. Thus, we can substitute the expression... 

The dissociation constant of acetic acid is 1.754 x 10. Calculate the degree of dissociation of 0.01-molar acid in the presence of 0.01-molar NaCl. Use the Debye-Hiickel limiting law to calculate the activity coefficients of the ions. Take the activity coefficient of the undissociated acid as unity. In your calculation neglect the concentration of in comparison with the concentration of Na+. 

For the solutions of Exercises 12 through 18, the Debye-Hiickel limiting law is sufficiently accurate, and the numbers in parentheses can be read as either molality or molarity at this level of approximation. The temperature to be used is 25°C. 

Calculate the mean ionic activity coefficient y+ of a 0.1 M water solution of NaCl at 25°C in water, using the Debye-Hiickel limiting law. 

The Debye Hiickel Limiting Law. Although beyond the scope of this Handbook, the derivation of this quantitative relationship rests on the following simplifying assumptions (a) Electrolytes are assumed to be completely... 

The Extended Debye-Hiickel Equation. This exercise reminds us that the Debye-Hiickel limiting law is not sufficiently accurate for most physicochemical studies. To estimate the calculated activity coefficient more accurately, one must consider the fact that ions are not point charges. To the contrary, ions are of finite size relative to the distance over which the ions interact electrostatically. This brings us to the extended Debye-Hiickel equation ... 

In solution thermodynamics, the concentration (C) of ions is replaced by their activity, a, where a = Cy and y is the activity coefficient that takes into account nonideal behavior due to ion-solvent and ion-ion interactions. The Debye-Hiickel limiting law predicts the relationship between the ionic strength of a solution and y for an ion of charge Z in dilute solutions ... 

B is a constant that depends on the properties of the solution, for example, on its dielectric constant, and on the temperature. For water at 25°C, B = 0.51 Ll/2 mol 1/2. The Debye-Hiickel limiting law applies only for solutions of low ionic strength, for example, below 0.01 M for 1 1 electrolytes, such as NaN03, and below 0.001 M for electrolytes of higher charge. 

As a reminder that the level of approximation in Equation (21) is the same as that of the Debye-Hiickel limiting law, the following example continues from this last result to the Debye-Hiickel expression for the mean ionic activity coefficient of an electrolyte solution. 

EXAMPLE 12.1 Debye-Huckel Expression for Ionic Activity Coefficients. The Debye-Hiickel limiting law attributes all of the nonideality of an electrolyte solution to electrostatic effects... 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

Table II. Positive Deviations from the Debye—Hiickel Limiting-Law Calculated from the Polarized Spheres Model. Contributions from...

Again we wish to emphasize that it is the modified moments (aWtU ) and (V5(D) Moi) arising from the use of Onsager s theory of the moments in liquids that are important. With reasonable values of internal refractive index, the moments in vacuum can be reduced. Furthermore, our theory does account for values of j greater than those predicted by the Debye-Hiickel limiting law if the ions can be assumed to have dipole characteristics. 

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