Apparent Debye-Hiickel

Discussion of the Gouy-Chapman (GC) Solution In Figure 4 the GC potential profile of Eq. [26] is compared to the Debye-Hiickel (Eq. [32]) and apparent Debye-Hiickel (Eq. [93], discussed later) potentials for two monovalent salt concentrations (0.01 and 0.1 M) for a surface with charge density a = 0.01 eolA. (The value for the surface... 

Bulk Model Apparent Debye-Hiickel Surface Charge Density... 

The apparent Debye-Hiickel (ADH) potential written in terms of the apparent Gouy-Chapman length is... 

By comparing the actual charge density to the apparent charge density in Fq. [94], we can define the fraction of surface charge neutralized by counterions, /iieutj according to the apparent Debye-Hiickel solution, as... 

The apparent Debye-Hiickel potential for a cylinder within the NLDH approximation is determined from Eqs. [188], [187], and [252]. We then have... 

The perturbed Gouy-Chapman expression of Eq. [152] may also be used to derive an apparent Debye-Hiickel potential similar to Eq. [308]. Using the approximate solution for az z electrolyte given by either Eqs. [147] and [149] or Eqs. [152] and [154], we can obtain an apparent DH potential far from a slightly curved charged surface. The two leading terms in the asymptotic expansion of Eq. [152] are... 

Plotting ixbase VS. pH gives a sigmoidal curve, whose inflection point reflects the apparent base-pAi, which may be corrected for ionic strength, I, using Equation 6.11 in order to obtain the thermodynamic pATa value in the respective solvent composition. Parameters A and B are Debye-Hiickel parameters, which are functions of temperature (T) and dielectric constant (e) of the solvent medium. For the buffers used, z = 1 for all ions ao expresses the distance of closest approach of the ions, that is, the sum of their effective radii in solution (solvated radii). Examples of the plots are shown in Figure 6.12. 

It is apparent from this plot that the Debye-Hiickel theory is only correct in a narrow range of extreme dilution, showing (as might have been anticipated) that classical electrostatics is qualitatively inadequate to describe the realistic interactions in aqueous electrolyte solutions, except in the asymptotic limit of extremely large ion-ion separations. 

An example not previously discussed is the Pitzer-Debye-Hiickel slope for apparent molar volume (Av) that is required in Eqs. 2.76, 2.80, and 2.81. A numerical equation for Ay as a function of temperature and pressure was derived from the database of Ananthaswamy and Atkinson (1984) over a temperature range of 273 to 298 K and over a pressure range of 1 to 1000 bars ... 

Fig. 3.25. The Pitzer-Debye-Hiickel slope for apparent molar volume (Av) as a function of temperature and pressure. Symbols are from Ananthaswamy and Atkinson (1984) lines represent model estimates...

We might proceed by plotting versus m, drawing a smooth curve through the points, and constructing tangents to the curve at the desired concentrations in order to measure the slopes. However, for solutions of simple electrolytes, it has been found that many apparent molar quantities such as tp vary linearly with yfm, even up to moderate concentrations. This behavior is in agreement with the prediction of the Debye-Hiickel theory for dilute solutions. Since... 

The interpretation of titration curves of peptides and proteins can be quite tricky. In addition to the number of groups that may be involved, their pAa values can be perturbed by several factors. For example, when charged groups are in close proximity and when salts are present, pA, values are influenced by electrostatic effects. Titration thus gives apparent pAa values and the intrinsic values have to be computed by applying a correction factor based on the Debye-Hiickel theory ... 

The theory of electrolyte solutions developed in this chapter relies heavily on the classical laws of electrostatics within the context of modern statistical mechanical methods. On the basis of Debye-Hiickel theory one understands how ion-ion interactions lead to the non-ideality of electrolyte solutions. Moreover, one is able to account quantitatively for the non-ideality when the solution is sufficiently dilute. This is precisely because ion-ion interactions are long range, and the ions can be treated as classical point charges when they are far apart. As the concentration of ions increases, their finite size becomes important and they are then described as point charges within hard spheres. It is only when ions come into contact that the problems with this picture become apparent. At this point one needs to add quantum-mechanical details to the description of the solution so that phenomena such as ion pairing can be understood in detail. 

The apparent close fit to the Debye-Hiickel-Onsager equations at low concentrations has to be reassessed in the light of the cross-over predicted by the later Fuoss-Onsager 1957 equation (see Section 12.10.2). 

After the Arrhenius theory was first proposed, an attempt was made to fit all conductance data to the Ostwald dilution law. It soon became apparent that many substances did not conform to this law. These substances are the strong electrolytes, which are completely dissociated into ions. The discussion of the dependence of the molar conductivity of strong electrolytes on concentration is based on the ideas contained in the Debye-Hiickel theory. 

Equations (3-6) for the potentiometric and spectrophotometric methods will provide thermodynamic pKa values. For the solubility-pH dependence method [Eqs. (7-8)], the values obtained are apparent values (pKg )/ which are relevant to the ionic strength (7) of the aqueous buffers used to fix the pH value for each solution. If the ionic strength of each buffer solution is controlled or assessed, then the apparent value can be corrected to a thermod)mamic value, using an activity coefficient from one of the Debye-Hiickel equations (Section 2.2.5). If the solubility-pH dependence is measured in several buffer systems, each with a different ionic strength, then the Guggenheim approach can be used to correct the result to zero ionic strength [Eq. (17)]. 

This 268 page article is concerned with the prediction of the thermodynamic properties of aqueous electrolyte solutions at high temperatures and pressures. There is an extensive discussion of the fundamental thermodynamics of. solutions and a discussion of theoretical concepts and models which have been used to describe electrolyte solutions. There is a very extensive bibliography ( 600 citations) which contains valuable references to specific systems of interest. Some specific tables of interest to this bibliography contain Debye-Hiickel parameters at 25 C, standard state partial molar entropies and heat capacities at 25 °C, and parameters for calculating activity coefficients, osmotic coefficients, relative apparent and partial molar enthalpies, heat capacities, and volumes at 25 °C. 

In the case of aqueous electrolytes the classical Debye-Hiickel theory (Debye and Hiickel, 1923) predicts that the excess apparent molar volume of the solute is given by... 

The standard electrode potential of an electrode is a very important electrochemical quantity. Conventionally, it is determined by the extrapolating the electrode potentials of extremely dilute solution along the line predicted by the Debye-Hiickel theory. Nevertheless, in the thermoelectrochemistry, it can be obtained by the measurement of the apparent enthalpy change. Based on a thermodynamic principle mentioned above (see Eq. (25)), Eq.(16) can be rewritten as... 

The molar volumes of ions in solution depend on their concentrations according to Equation 2.25 as discussed in Section 2.3.1.5 for the apparent molar volumes. This expression involves the limiting slope of the square root of the concentration, which is according to the Debye-Hiickel theory [4,26] ... 

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