Debye-HOckel

CHEMICAL THERMODYNAMICS DEBYE-HOCKEL MODEL GIBBS FREE ENERGY... 

COULOMB COULOMB S LAW DEBYE-HOCKEL TREATMENT COULOMETRIC TITRATION 4-COUMAROYL-CoA SYNTHETASE COUNTERION Counting efficiency,... 

DAVIES EQUATION DEBYE-HOCKEL TREATMENT Day-night cycle,... 

BIOMINERALIZATION DEBYE-HOCKEL TREATMENT DEFINITE INTEGRAL Degenerate rearrangement,... 

Debye-HOckel screening parameter, Eq. (43) common logarithm of the retention factor... 

At higher ionic strength the limiting Debye-HOckel treatment no longer adequately describes the effect of the ionic atmosphere on the activity of an ion, and this deficiency has been the subject of much theoretical investigation. 

Two semiempirical approaches which to date seem to account best for the effects observed at high ionic istrength, will be discussed here. The first, due to Lietzke et at. 197), views the solution as a mixture of two components one has the limiting Debye-HOckel character apd the other component exhibits the behavior of fused salts. The activity coefficient for fused salts is given by... 

By measuring the solubility, r, of the silver chloride in different concentration of added salt and extrapolating the solubilities to zero salt concentration, or better, to zero ionic strength, one obtains the solubility when v = 1. and from Eq. (29) K can be found. Then y can be calculated using this value of K and any measured solubility. Actually, this method is only applicable to sparingly soluble salts. Activity coefficients of ions and of electrolytes can be calculated from the Debye-HOckel equations. For a uni-univalent electrolyte, in water at 25 C, the equation for the activity coefficient of an electrolyte is... 

DEBYE-HOCKEL LIMITING LAW. The departure from ideal behavior in a given solvent is governed by the ionic strength of the medium and the valences of the ions of the electrolyte, but is independent of their chemical nature. For dilute solutions, the logarithm of the mean activity is proportional to the product of the cation valence, anion valence, and square root of ionic strength giving the equation... 

TABLE XXXV. debye-hOckel constants AND DIELECTRIC CONSTANT OF ... 

For spherical symmetry the Polsson-Boltzmann equation cannot be solved analytically, but numerical solutions are nowadays available. Analytical solutions exist for low potentials, that is In the Debye-HOckel (DH) approximation, already encountered in the treatment of the ionic atmosphere around Ions, sec. 1.5.2a. As compared with flat double layers, the low-potential approximation tends to become better, the smaller the particle is. The reason is that, because of the stronger divergence of the lines of force, the potential decays more rapidly a relatively larger fraction of the countercharge is therefore found in the region of low potentials. 

Figure 3.12. Charge in a spherical diffuse double layer. Electrolyte (1-1), 10 M. Temperature 25°C. The particle radius a is given. Dashed curves Debye-HOckel approximation.

The first attempt in this direction dates back to the Just-mentioned Smolu-chowski theorem. For the development of electrophoresis theory a very important contribution was made by Henryk) who systematically studied the distortion of the field by spherical and cylindrical particles. It depends on the size and shape of the particle, and on K /K as illustrated in fig. 3.84. The electrophoretic friction that is created also depends on x. For this type of theory a double layer picture is needed, but because of mathematical difficulties Henry had to limit himself to the linearized (Debye-HOckel) approximation. As a consequence, Henry s results are only valid for low Notwithstanding this... 

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

The Hiickel equation (41.13), appropriately adjusted to give 7m, has been frequently employed for the analytical representation of activity coefficient values as a function of the ionic strength of the solution, and various forms of the Debye-HOckel and Br nsted equations have been used for the purpose of extrapolating experimental results. Some instances of such applications have been given earlier ( 39h, 39i), and another is described in the next section. 

Table II. Standard Potential of the Cell, Zn ZnS04l PbS04(s) Pb(Hg) at 25° as Computed by the Gronwall, LaMer and Sandved Extension of the Debye-HOckel Theory...

Figure 1. Experimental and calculated solubiliti ofSr(OH) 8H20 in NaOK Patterned line represents calculations with Sr OH interactions described solely with the use ofPitzer s form of the extended Debye-HOckel equation. Solid line represents the calculations of our final thermodynamic model, which includes values for the Pitzer ion interaction parameters. Total concentrations in units of molarity. From (3).

One must remember that this common language can be confusing. In particular, the activity of an ion can depend a priori on the value of the electric potential. By way of simplifying, the so-called chemical potential is often said not to depend on the electric potential but rather depend exclusively on the medium s chemical composition (as in the Debye-HOckel law, see section 3.2.1.3). 

Let us specify the different hypotheses laid out by the Debye-HOckel model and the subsequent approximations which help one develop a means of expressing the activity coefficient of various ions in an electrolyte. Nearly all the assumptions, expressed below, are based on the fact that this theory is developed for dilute solutions ... 

AppendixA.3.3 lays out the full reasoning and calculations behind Debye-HOckel s limiting law, covering all the various steps, based on the assumptions listed here. 

In reality, these experimental results are not wholly inconsistent with the Debye-HOckel model, whereby the mean activity coefficient decreases as the molality increases. This comes down to taking into account the fact that a usual solute/solvent description is no longer satisfactory for concentrated electrolytes. Indeed, when dealing with concentrated electrolytes, the number of solvent molecules involved in the solvation sphere, close to the ions, cannot be ignored when compared to the total number of solvent molecules. Once one takes this phenomenon into account, then three types of adjustment emerge, each of which are laid out in detail below. 

The extended Debye-HOckel law is described in more detail in appendixA.3.3. 

Also shown here is the curve calculated from the extended Debye-HOckel law, having taken into account the experimental density variations In the concentrated solutions (i.e., the relationship between molality and concentration). This figure shows how, if one applies the correct terms to define concentration in highly concentrated media, then the experimental mean activity coefficients are not too far off those prescribed by the extended Debye-HOckel law. There Is no minimum on the curve with actual values, and the mean activity coefficient never exceeds 1. 

The Debye length is defined in mathematical terms In appendix A.3.3 which describes the Debye-HOckel theory. Some orders of magnitude are given later in the text. 

Assuming that the cation and anion activity coefficients are equal, which is in the case particular in the context of the Debye-HOckel theory, simplifies the calculation yet without making the reasoning any less general. 

Contrary to the hypothesis made in the Debye-HOckel model (see appendix A.3.2). 

Here we will briefly summarise the basic elements that are needed for demonstrating the Debye-HOckel laws as a means of estimating the ionic activity coefficients in a solution. 

When the ionic strength is higher than 10 mol L" then the following extended Debye-HOckel equation can be used ... 

Based on the simplified Debye-HOckel model, if you take the mean activity coefficient of a solute in a solution containing only NaCI, and compare it to the mean activity coefficient in a solution with the same ionic strength containing only Cu(NC>3)2 then the former coefficient is... 

---