Debye—Huckel charging function, equation

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... 

The Debye-Huckel theory was developed to extend the capacitor model and is based on a simplified solution of the Poisson equation. It assumes that the double layer is really a diffuse cloud in which the potential is not a discontinuous function. Again, the interest is in deriving an expression for the electrical potential function. This model states that there is an exponential relationship between the charge and the potential. The distribution of the potential is ... 

Understandably, it is much more common to see analyses of problems based on Eq. (32) since for simple geometries the solution can be written down in closed form, expressed in terms of simple functions. For plane surfaces, for example, the solutions are elementary hyperbolic functions while for an isolated spherical surface the Debye-Huckel potential expression prevails. For two charged spherical surfaces the general solution can be written down as a convergent infinite series of Legendre polynomials [16-19]. The series is normally truncated for calculation purposes [16] K For an ellipsoidal body ellipsoidal harmonics are the natural choice for a series representation [20]. (The nonlinear Poisson Boltzmann equation has been solved numerically for a ellipsoidal body... 

Here we treat a planar plate surface immersed in an electrolyte solution of relative permittivity e,. and Debye-Huckel parameter k. We take x- and y-axes parallel to the plate surface and the z-axis perpendicular to the plate surface with its origin at the plate surface so that the region z>0 corresponds to the solution phase (Fig. 2.1). First we assume that the surface charge density a varies in the x-direction so that a is a function of x, that is, cr = cr(x). The electric potential ij/ is thus a function of x and z. We assume that the potential i/ (x, z) satisfies the following two-dimensional linearized Poison-Boltzmann equation, namely,... 

As an illustrative example taken from Russel et al. (1989), let us consider a 0.01 molar solution of sodium chloride in contact with a surface charged at a density of 5 x 10 negative charges per square meter at room temperature, 298°K. Equation (2-52) gives /c = 3 nm. The dimensionless surface potential exfJkrtT. obtained from Eq. (2-45), is —5.21, and Eqs. (2-46) and (2-49) give respectively the exact and the Debye-Huckel approximations for the potential as a function of distance from the surface. The results are plotted in Fig. 2-13. Note that since — s/ ks T > 1, the Debye-Huckel approximation is... 

Some ion activity coefficients at 25°C computed with the Debye-Huckel equation as a function of ionic strength, ion size, and charge, are shown in Table 4.2. Debye-Huckel ion activity coefficients up to 0.1 mol/kg ionic strength, are plotted in Fig. 4.3 for some monovalent and divalent ions. The Debye-Huckel equation can be used to compute accurate activity coefficients for monovalent ions up to about / = 0.1 mol/kg, for divalent ions to about / = 0.01 mol/kg, and for trivalent ions up to perhaps / = 0.001 mol/kg. 

Attempts to improve the theory by solving the Poisson-Boltzmann equation present other difficulties first pointed out by Onsager (1933) one consequence of this is that the pair distribution functions g (r) and g (r) calculated for unsymmetrically charged electrolytes (e.g., LaCl or CaCl2) are not equal as they should be from their definitions. Recently Outhwaite (1975) and others have devised modifications to the Poisson-Boltzmann equation which make the equations self-consistent and more accurate, but the labor involved in solving them and their restriction to the primitive model electrolyte are drawbacks to the formulation of a comprehensive theory along these lines. The Poisson-Boltzmann equation, however, has found wide applicability in the theory of polyelectrolytes, colloids, and the electrical double-layer. Mou (1981) has derived a Debye-Huckel-like theory for a system of ions and point dipoles the results are similar but for the presence of a... 

The most important parameter in equations (7.37) to (7.39) is k, which has the dimension of the reciprocal of length. In water at 25 C, = 0.329 /7A". Distance is the Debye-Huckel length and represents the thickness of the diffuse layer. This happens to be a misnomer because, over distance /c , the potential decreases only by j/exp(l) = tpd/2.1, but, in the weak potential approximation, the diffuse layer [equation (7.38)] can be treated as a parallel-plate capacitor Q = 6K with plates separated by distance /c . The variation in the potential in the solution, as a function of the distance from the surface, depends on the concentration and the charge of the ions present in the electrolyte (Figure 7.6). 

---