Debye-Huckel variation

Equation 6-33 suggests that extrapolation of equilibrium constants to infinite dilution is done appropriately by plotting log Kc vs-01. For example, Fig. 6-1 shows plots of pK a for dissociation of H2P04-, AMP, and ADP2-, and ATP3 vs fp. The variation of pK a with- p at low concentrations (Eq. 6-35) is derived by application of the Debye-Huckel equation (Eq. 6-33) ... 

Taking the surface potential to be xp°, the potential at a distance x as rp, and combining the Boltzmann distribution of concentrations of ions in terms of potential, the charge density at each potential in terms of the concentration of ions, and the Poisson equation describing the variation in potential with distance, yields the Pois-son-Boltzmann equation. Given the physical boundary conditions, assuming low surface potentials, and using the Debye-Huckel approximation, yields... 

To determine the spatial variation of a static electric field, one has to solve the Poisson equation for the appropriate charge distribution, subject to such boundary conditions as may pertain. The Poisson equation plays a central role in the Gouy-Chapman (- Gouy, - Chapman) electrical - double layer model and in the - Debye-Huckel theory of electrolyte solutions. In the first case the one-dimensional form of Eq. (2)... 

I should note that the limit of sensitivity of these experiments restricts us to saying that the phase transitions are simultaneous only in the sense that they both occur between -1 and 0°C (in either direction). More subtle variations, of the order of 0.1°C, would not have been detectable. According to Debye-Huckel theory [3], the depression of the freezing point of pure water is 0.18°C in a 0.1 M uni-univalent electrolyte solution. We would expect the clay to cause a further small depression in the freezing point, as discussed below. Within these limits, the temperature where both the freezing transition and the gel-crystalline phase transition occur is the same in our model clay colloid system, and it can be concluded to be the ordinary freezing point of the soaking solution. 

In addition to the short-range interactions between species in all solutions, long-range electrostatic interactions are found in electrolyte solutions. The deviation from ideal solution behavior caused by these electrostatic forces is usually calculated by some variation of the Debye-Huckel theory or the mean spherical approximation (MSA). These theories do not include terms for the short-range attractive and repulsive forces in the mixtures and are therefore usually combined with activity coefficient models or equations of state in order to describe the properties of electrolyte solutions. 

The various forms of equation (40.15), referred to as the Debye-HUckel limiting law, express the variation of the mean ionic activity coefficient of a solute with the ionic strength of the medium. It is called the limiting law because the approximations and assumptions made in its derivation are strictly applicable only at infinite dilution. The Debye-Hfickel equation thus represents the behavior to which a solution of an electrolyte should approach as its concentration is diminished. 

A typical disjoining pressure variation with film thickness is shown in Figure 7a. At small thicknesses, a repulsive force is observed which can be fitted with an exponential form exp-(jch), as expected for screened electrostatic repulsion k is close to the calculated inverse Debye Huckel length in the solution ... 

Figure 12.12. Variation of the diffusion coefficient as a function of the distance (p) from the center of the pore (i.e. p = 0 A) to within 1 A of the -SOs groups. The calculation is for Nafion with 6 H2O/ SOs" with inclusion of either Debye-Huckel or Attard models for the shielding. At the center of the pore the Attard formulation results in the most significant screening of the sulfonate groups and hence the higher proton self-diffusion coefficient.

The interaction between ions, mainly between those of opposite charge, can lead to a variation of activity coefficients with the composition of mixed electrolyte solutions and thus contradict the Debye-Huckel formula prediction that log ft is proportional to In dilute solutions the conflict is minimal, but at higher concentrations (>0.1 mol dm ) the influence of ion interactions can, for example, alter the significance of the ionic size parameter, d, in the flrst term of Equations (12) and (13), namely ... 

The separation of Tp into ratios for the solution and solid phases is useful when solution phase activity coefficients are available (5) because it allows evaluation of variations from ideality of the clay phase alone over a range of experimental conditions. When measurements for the aqueous mixed electrolyte systems in question are not available, adequate estimates can frequently be made by Debye-Huckel equations or by various methods from measurements on two-component systems (6). 

For further simplifications, one may note that for small values of the argument, it is possible to neglect higher order terms in an exponential series and approximate exp(jc) as I+x. This consideration, when applied to the potential distributions depicted by Eqs. (12a), (12b), forms the basis of the Debye-Huckel linearization principle [2], which effectively linearizes the pertinent exponential variation of ionic charge distribution for small values of e /k T. Under such approximations, Eq. (13) can be simplified as... 

Fig. 3.24 Plot of the structure factor for a weak polyelectrolyte with A = 0.1. Part (a) is for Coulomb interactions and part (b) is for Debye-Huckel interactions with ko = 0.05. The dotted line has slope -1, and the points are from a variational theory. (Taken from Ref. 153).

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). 

These philosophies have been the source of some of the choices made when balancing what should be retained and what rewritten. The result is quite heterogeneous. Chapters 1 and 2 are completely new. The contributions from neutron diffraction measurements in solutions and those from other spectroscopic methods have torn away many of the veils covering knowledge of the first 1-2 layers of solvent around an ion. Chapter 3 also contains much new material. Debye and Huckel s famous calculation is two generations old and it is surely time to move toward new ideas. Chapter 4, on the other hand, presents much material on transport that is phenomenological—material so basic that it must be presented but shows little variation with time. 

Figure 4.3 shows a comparison of the variations of the mean activity coefficient of magnesium chloride as a function of the ionic strength. The points are obtained experimentally and the downward curve is obtained by application of Debye and HiickeTs law. The figure demonstrates that the model begins to deviate from the experimental values long before the ionic strength reaches one. In particular, for most electrolytes, the real curve exhibits an extremum which Debye and Huckel s model never shows. 

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