Debye-Htickel model

When the Debye-Htickel model was developed, an important hypothesis was made for mathematical convenience, i.e., yi, < kT. Now, using an aqueous 1 1 electrolyte solution at 25 °C as an example, reassess the validity of the hypothesis. What is the physical nature of the hypothesis (Xu)... 

For each model system developed in this book, make it a habit to write out the systems description whenever you encounter that model. Tliis includes the kinetic theory of gases, thermodynamic systems, the Born model, the Debye-Htickel model, electric circuit models of electrochemical systems, etc. 

The Debye-Htickel model considered the solvent to be a structureless medium whose only property is to reduce the interactions between ions in a vacuum by a factor given by the macroscopic relative permittivity, e. No cognisance was taken of the possibility of ion-solvent interactions, and the size of the ion was assumed to be that of the bare ion. Gurney in the 1930s introduced the concept of the co-sphere and this has proved to be a useful concept in the theory of electrolyte solutions. Many recent theories of conductance are based on the Gurney co-sphere concept (see Section 12.17). 

In solvents with high dielectric constant, such as water, the solution deviates very little from ideal solutions if they are diluted enough (about 0.1 M) and can be treated using the Debye-Htickel model, as described in section 11.4. 

The Debye-Htickel ionic-cloud model for the distribution of ions in an electrolytic solution has permitted the theoretical calculation of the chemical-potential change arising from ion-ion interactions. How is this theoretical expression to be checked, i.e., connected with a measured quantity It is to this testing of the Debye-Huckel theory that attention will now be turned. 

The ion size parameter a has done part of the job of extending the range of concentration in which the Debye-Htickel theory of ionic clouds agrees with experiment. Has it done the whole job One must start looking for discrepancies between theory and fact and for the less satisfactory features of the model. 

This does not mean that the Debye-Htickel theory gives the right answer when there is ion-pair formation. The extent of ion-pair formation decides the value of the concentration to be used in the ionic-cloud model. By removing a fraction 0 of the total number of ions, only a fraction 1 - 0 of the ions remain for the Debye-Hiickel treatment, which interests itself only in the free charges. Thus, the Debye-Htickel expression for the activity coefficient [Eq. (3.120)] is valid for the free ions, with two important modifications (1) Instead of there being a concentration c of ions, there is only (1 - 0)c the remainder Oc is not reckoned with owing to association. (2) The distance of closest approach of free ions is q and not a. These modifications yield... 

Why should one go to all this trouble and do all these integrations if there are other, less complex methods available to theorize about ionic solutions The reason is that the correlation function method is open-ended. The equations by which one goes from the gs to properties are not under suspicion. There are no model assumptions in the experimental determination of the g s. This contrasts with the Debye-Htickel theory (limited by the absence of repulsive forces), with Mayer s theory (no misty closure procedures), and even with MD (with its pair potential used as approximations to reality). The correlation function approach can be also used to test any theory in the future because all theories can be made to give g(r) and thereafter, as shown, the properties of ionic solutions. 

The correction factor,/, relates the actual mobility of a fully charged particle at the ionic strength under the experimental conditions to the absolute mobility. It takes ionic interactions into account and is derived for not-too-concentrated solutions by the theory of Debye-Htickel-Onsager using the model of an ionic cloud around a given central ion. It depends, in a com-... 

For low pressures (a few atmospheres and lower) we can apply the ideal gas model for gases and ideal mixture models for liquids. This formulation is very common in reactor technology. In some cases at higher pressures, the pressure effect on the gas phase is important. A suitable model for these systems is to use an EOS for the gas phase, and an ideal mixture model for liquids. However, in most situations at low pressures the liquid phase is more non-ideal than the gas phases. Then we will rather apply the ideal gas law for the gas phase, and excess properties for liquid mixtures. For polar mixtures at low to moderate pressures we may apply a suitable EOS for gas phases, and excess properties for liquid mixtures. All common models for excess properties are independent of pressure, and cannot be used at higher pressures. The pressure effect on the ideal (model part of the) mixture can be taken into account by the well known Poynting factor. At very high pressures we may apply proper EOS formulations for both gas and liquid mixtures, as the EOS formulations in principle are valid for all pressures. For non-volatile electrol3d es, we have to apply a suitable EOS for gas phases and excess properties for liquid mixtures. For such liquid systems a separate term is often added in the basic model to account for the effects of ions. For very dilute solutions the Debye-Htickel law may hold. For many electrolyte systems we can apply the ideal gas law for the gas phase, as the accuracy reflected by the liquid phase models is low. 

FIG. 18 Zeta potential = //(/, + cr/2) as a function of Manning parameter for the DNA-like models with 0.5 mol/L added 2 2 salt. The dotted line is the prediction of PB theory, the dash-dotted line is from bulk Debye—Htickel theory, and the dashed line is the result from a hypernetted chain calculation [36]. The solid line is a fit that merely serves to guide the eye. 

The model and theory, like that of the Debye-Htickel treatment of non-ideality, were based on consideration of long range electrostatic coulombic interactions only. The model was most likely to be inadequate because it did not take into account specific short range interactions corresponding to ion-ion, ion-solvent, and solvent-solvent interactions. 

The specific interaction parameter b is also different from the corresponding Ae value because of the different Debye-Htickel terms. Up to the review of Baes and Mesmer [1976BAE/MES], there were no data available to model the ionic strength dependence of the equilibrium constants for polynuclear species in one of the ionic media. The common method to estimate equilibrium constants at zero ionic strength was to use an experimental value at finite ionic strength and then to use only the Debye-Htickel term to estimate the value at zero ionic strength. 

The ionic strength dependence of Reaction (A.94) is very large the values of logjo K ail = 0.025, 0.050 and 0.70 are -9.3, -7.7 and -7.0, respectively this is not what one expects from the SIT model where the ionic strength effect in this case is determined mainly by the Debye-Htickel term. 

The restricted primitive model described at the beginning of Sect. 2.2. regards the ions as charged conducting spheres dispersed uniformly in a continuum made up of a compressible dielectric. Within this model and in very dilute solutions of electrolytes the well-known Debye-Htickel theory describes the chemical potentials of the electrolyte, /xe, and that of the water, /u-w, sufficiently well. 

The model relates the excess total Gibbs energy of a system to an equation similar in principle to the virial equation we saw for gases in 13.5, in which the first term is not the ideal gas expression, but a simplified form of the Debye-Htickel equation. The general equation used by Pitzer (from which many others are derived by differentiation) is... 

The first virial coefficient /(/) is some function of the ionic strength and is not 0 as it would be for an ideal solution, but is in fact a version of the Debye-Htickel equation, which represents departure from ideality in very dilute solutions. The following term is a function of the interactions of all pairs of ions, and the third term a function of the interactions of ions taken three at a time. The second coefficient. Ay, is a function of ionic strength, but the third coefficient ju-y - is considered to be independent of ionic strength and equals zero if /, j, and k are all anions or cations. Later extensions to the model published by Pitzer and co-workers allow for an ionic strength dependence to the third coefficient. Pitzer (1987) and Harvie and Weare (1980) note that higher virial coefficients are required only for extremely concentrated solutions, so the series is stopped at the third coefficient. 

Chemical equilibrium in a closed system at constant temperature and pressure is achieved at the minimum of the total Gibbs energy, min(G) constrained by material-balance and electro-neutrality conditions. For aqueous electrolyte solutions, we require activity coefficients for all species in the mixture. Well-established models, e.g. Debye-Htickel, extended Debye-Hiickel, Pitzer, and the Harvie-Weare modification of Pitzer s activity coefficient model, are used to take into account ionic interactions in natural systems [15-20]. 

The electrostatic correlations of the ions in the electrolyte solution contribute to the Helmholtz free energy of the system, depending on the Debye length and ion size. Based on the Debye-Htickel theory for the restricted primitive model (Figure 3.3a), the result (McQuarrie 1976) is... 

SASA), a concept introduced by Lee and Richards [9], and the electrostatic free energy contribution on the basis of the Poisson-Boltzmann (PB) equation of macroscopic electrostatics, an idea that goes back to Born [10], Debye and Htickel [11], Kirkwood [12], and Onsager [13]. The combination of these two approximations forms the SASA/PB implicit solvent model. In the next section we analyze the microscopic significance of the nonpolar and electrostatic free energy contributions and describe the SASA/PB implicit solvent model. 

Debye and Htickel s model is presented in Chapter 4 of Volume 2 in this series of books [SOU 15b],... 

A third model, given by Debye and Htickel [GOK 96a], is also used for diluted solutions of charged components (ions, structure elements). 

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