Debye-Huckel equation limiting

Fuoss and Kraus [13] and Shedlovsky [14] improved Eq. (7.6) by taking the effect of ion-ion interactions on molar conductivities into account. Here, Fuoss and Kraus used the Debye-Huckel-Onsager limiting law [Eq. (7.1)] and Shedlovsky used the following semi-empirical equation ... 

Abstract, The solution of the Debye-Huckel equation for a system of spheres with arbitrary radii and surface charge in electrolyte solutions is described. The general theoretical approach to describe such systems is elaborated. The practically important case of two spheres is considered in detail. Finite closed formulae to calculate the interaction energy of two spherical particles with constant surface charges are obtained from general expressions in zero approximation. Known relationships follow from our formulae in limiting cases. 

In water187), however, the slopes of the solute concentrations. This is expected as a limiting positive slope is predicted by the Debye-Huckel equation for all salts in all solvents189). Attempts to demonstrate a change to positive slope for the larger tetraalkylammonium iodides in NMA have not been successful186). These attempts used a method of limited precision and cannot be considered to be conclusive. 

FIGURE 2-3 Activity coefficients calculated by the limiting Debye-Huckel equation (dotted lines) and those observed experimentally. Left, electrolytes of three charge types in water. Right, hydrochloric acid in water-dioxane mixtures with bulk dielectric constants as indicated. Adapted from Homed and Ow . )... 

DERIVATION OF THE LIMITING FORM FOR THE DEBYE-HUCKEL EQUATION... 

In view of these uncertainties, it may prove more advantageous provisionally to work with an empirical relationship between activity coefficients and ionic strength of more concentrated solutions, instead of the Debye-Huckel equation. Bjerrum has found from experience that the following equation holds within wide limits ... 

The experimental methods of these authors are described in Chapter 22. For very dilute solutions the second term in the denominator of equation (25) becomes negligible and the Debye-Huckel equation approaches its limiting form,... 

For electrolyte solutions that are sufficiently dilute, the activity coefficient can be calculated from the limiting Debye-Huckel equation... 

The extended Debye-Huckel equation 12-5 predicts that the activity coefficient, 7, will decrease as ionic strength, x, increases. In fact, above an ionic strength of approximately 1 M, activity coefficients of most ions increase, as shown for in NaC104 solutions in Figure 12-5. We should not be too surprised that activity coefficients in concentrated salt solutions are not the same as those in dilute aqueous solution. The solvent is no longer just H2O but, rather, a mixture of H2O and NaC104. Hereafter, we limit our attention to dilute aqueous solutions in which Equation 12-5 applies. 

Figure 10.4 Dependence of In y on -Jhn for aqueous HCl (upper curves) and aqueous CaCl2 (lower curves) at 25 C." Solid curves experimental dashed curves Debye-Huckel equation (a = 5 x 10 m for HCl, a = 4.5 x 10 m for CaCb) dotted lines Debye-Huckel limiting law.

Derivation of the Limiting Form for the Debye-Huckel Equation... 

One other relevant theoretical set of work concerns the counterion distribution particularly in the dilute limit. Manning solved the Debye-HUckel equation for a single infinitely thin polyelectrolyte. He found that when a < Xb the counterions condense onto the hne polymer reducing the charge density until the charge separation becomes equal to the Bjerrum length. The details are altered when the Poisson-Boltzmann approximation is used for a cylindrical polyelectrolyte, " but the basic point of condensation occuring for A > 1 remains. In a similar vein, Oosawa proposed a two-phase model of bound and free counterions. These results are especially relevant, since many prototypical polyelectrolytes, such as DNA and NaPSS, have A 3,... 

Limiting and extended forms of the Debye-Huckel equation... 

Figure 7.8 Comparison of experimental ln7 for 1 1, 2 1, and 2 2 electrolytes. The symbols indicate the experimental results, with representing HC1 (z+ = 1, z = — 1) representing SrC ( + = 2, r = — 1) and A representing ZnS04 (z+ = 2, z = -2). The lines are the Debye-Huckel predictions, with the solid line giving the prediction for (z+ = 1, z = -1) the dashed line for (z+ = 2, r = -1) and the dashed-dotted line for (z+= 2, z =-2). In (a), In 7- calculated from the limiting law [equation (7.45)] is shown graphed against I 2. In (b). In 7- calculated from the extended form [equation (7.43)] is shown graphed against 7m2.

Further simphfication of the SPM and RPM is to assume the ions are point charges with no hard-core correlations, i.e., du = 0. This is called the Debye-Huckel (DH) level of treatment, and an early Nobel prize was awarded to the theory of electrolytes in the infinite-dilution limit [31]. This model can capture the long-range electrostatic interactions and is expected to be valid only for dilute solutions. An analytical solution is available by solving the Pois-son-Boltzmann (PB) equation for the distribution of ions (charges). The PB equation is... 

In this one-dimensional flat case the Laplace operator is simpler than in the case with spherical symmetry arising when deriving the Debye-Huckel limiting law. Therefore, the differential equation (B.5) can be solved without the simplification (of replacing the exponential factors by two terms of their series expansion) that would reduce its accuracy. We shall employ the mathematical identity... 

In fact, the symbol Ic should be used, as the molality ionic strength Im can be defined analogously in dilute aqueous solutions, however, values of c and m, and thus also Ic and Im, become identical.) Equation (1.1.21) was later derived theoretically and is called the Debye-Huckel limiting law. It will be discussed in greater detail in Section 1.3.1. 

The derivation of the equations of the Debye-Huckel theory did not require differentiation between a solution of a single electrolyte and an electrolyte mixture provided that the limiting law approximation Eq. (1.3.24), was used, which does not contain any specific ionic parameter. If, however, approximation (1.3.29) is to be used, containing the effective ionic diameter ay it must be recalled that this quantity was introduced as the minimal mean distance of approach of both positive and negative ions to the central ion. Thus, this quantity a is in a certain sense an average of effects of all the ions but, at the same time, a characteristic value for the given central... 

Friedman (1962) has used the cluster theory of Mayer (1950) to derive equations which give the thermodynamic properties of electrolyte solutions as the sum of convergent series. The first term in these series is identical to and thus confirms the Debye-Huckel limiting law. The second term is an I2.nl term whose coefficient is, like the coefficient in the Debye-Huckel limiting law equation, a function of the charge type of the salt and the properties of the solvent. From this theory, as well as from others referred to above, a higher order limiting law can be written as... 

In single electrolyte solutions the Debye-Huckel limiting equation works up to 10-3 m. Its well known extended form used at m > 10"3 is... 

The only three methods which do not require curve-fitting at present are the use of the Debye-Huckel limiting law and of the equations of Guntelberg and of Davies. Unfortunately these equations are of value only in very dilute and simple solutions. 

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