Debye-Hiickel model

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

Bearing in mind these limitations on the Debye—Hiickel model of electrolytes, the influence of ionic concentration on the rate coefficient for reaction of ions was solved numerically by Logan [54, 93] who evaluated the integral of eqn. (56) with the potential of eqn. (55). He compared these numerical values with the predictions of the Bronsted— Bjerrum correction to the rate of a reaction occurring between ions surrounded by equilibrated ionic atmospheres, where the reaction of encounter pairs is rate-limiting... 

Issue is taken here, not with the mathematical treatment of the Debye-Hiickel model but rather with the underlying assumptions on which it is based. Friedman (58) has been concerned with extending the primitive model of electrolytes, and recently Wu and Friedman (159) have shown that not only are there theoretical objections to the Debye-Hiickel theory, but present experimental evidence also points to shortcomings in the theory. Thus, Wu and Friedman emphasize that since the dielectric constant and relative temperature coefficient of the dielectric constant differ by only 0.4 and 0.8% respectively for D O and H20, the thermodynamic results based on the Debye-Hiickel theory should be similar for salt solutions in these two solvents. Experimentally, the excess entropies in D >0 are far greater than in ordinary water and indeed are approximately linearly proportional to the aquamolality of the salts. In this connection, see also Ref. 129. 

Figure 2 shows y for a few electrolytes as a function of m, as extracted from Robinson and Stokes (2002), as well as their calculated values using the Debye-Hiickel model integrated with a linear function of m to extend its accuracy beyond the dilute region ... 

Correlations for the determination of the dissociation equilibrium constants and solubility values for SO2 and CO2 as functions of temperature as well as the equations for activity coefficients are given in Ref. [70], Thermodynamic non-idealities are taken into account depending on whether species are charged, or not. For uncharged species, a simple relationship from Ref. [102] is applied, whereas for individual ions, the extended Debye-Hiickel model is used according to Ref. [103]. 

Hence, the complicated problem of the time-averaged distribution of ions inside an electrolytic solution reduces, in the Debye-Hiickel model, to the mathematically simpler problem of finding out how the excess charge density q varies with distance r from the central ion. 

Actually, there are discrete charges in the neighborhood of the central ion and therefore discontinuous variations in the potential. But because in the Debye-Hiickel model the charges are smoothed out, the potential is averaged out. 

Before the activity coefficients calculated on the basis of the Debye-Hiickel model can be compared with experiment, there arises a problem similar to one faced in the discussion of ion-solvent interactions (Chapter 2). Thae, it was realized the heat of hydration of an individual ionic species could not he measured because such a measurement would involve the transfer of ions of only one species into a solvent instead of ions of two species with equal and opposite charges. Even if such a transfer were physically possible, it would result in a charged solution and therefore an extra, undesired interaction between the ions and the electrified solution. The only way out was to transfer a neutral electrolyte (an equal number of positive and negative ions) into the solvent, but this meant that one could only measure the heat of interactions of a salt with the solvent and this experimental quantity could not be separated into the individual ionic heats of hydration. 

Eq. (3.60)] to the experimentally accessible mean ionic-activity coefficient/ so that the Debye-Hiickel model can be tested. 

The result of equating these two expressions for the excess charge density is the fundamental partial differential equation of the Debye-Hiickel model, the linearized P-B equation (Fig. 3.36),... 

Unfortunately, the value ofa obtained from experiment by Eq. (3.119) varies with concentration (as it would not if it represented simply the collisional diameters), and as the concentration increases beyond about 0.1 moldm, a sometimes has to assume physically impossible (e.g., negative) values. Evidently these changes demanded by experiment not only reflect real changes in the sizes of ions but represent other effects neglected in the simplifying Debye-Hiickel model. Hence, the basic postulates of the Debye-Hiickel model must be scrutinized. 

One approach to understanding the discrepancies between the experimental values of the activity coefficient and the predictions of the Debye-Hiickel model has just been described (Section 3.6) it involved a consideration of the influence of solvation. 

Using the point-charge version of the Debye-Hiickel model, derive the radial distribution of total excessive charge q r) from the central ion. Comment on the difference between q r) and the excessive charge in a dr-thickness shell dq r). (Xu)... 

The linearization that leads here to the Debye-Hiickel model is physically consistent in this argument. But the possibility of a model that is unlinearized in this sense is a popular query. More than one response has been offered including the (nonlinear) Poisson-Boltzmann theory and the EXP approximation see (Stell, 1977) also for representative numerical results for the systems discussed here. 

Q.l5.7 What assumption of the Debye-Hiickel model permits the use of the linearized Boltzmann equation Is this assumption valid if ZiCo r/kT = 0.5 0.1 0.01 ... 

Q.l5.9 Is the Debye-Hiickel model a good model for ionic interactions within a biological system ... 

A.15.9 The Debye-Hiickel limiting law suggests it is not a good model because the ionic strength of solutions in vivo are too high to be accurately modeled by Debye-Hiickel model. 

In the present chapter, the properties of electrolyte solutions in water are discussed in detail. Initially the solvation of ions in infinitely dilute solutions is considered on the basis of the Born theory. Then, the Debye-Hiickel model for... 

This is exactly the form expected for gyir) on the basis of the Debye-Hiickel model if 2F replaces k as the screening parameter. 

Fig. 3.8 Plot of the mean ionic activity coefficient for NaCl at 25°C on a logarithmic scale against the square root of the molar ionic strength. The points give the experimental data the curves give the theoretical fits for the MSA and extended Debye-Hiickel models with ion size parameters as indicated.

In conclusion, the MSA provides an excellent description of the properties of electrolyte solutions up to quite high concentrations. In dilute solutions, the most important feature of these systems is the influence of ion-ion interactions, which account for almost all of the departure from ideality. In this concentration region, the MSA theory does not differ significantly from the Debye-Hiickel model. As the ionic strength increases beyond 0.1 M, the finite size of all of the ions must be considered. This is done in the MSA on the basis of the hard-sphere contribution. Further improvement in the model comes from considering the presence of ion pairing and by using the actual dielectric permittivity of the solution rather than that of the pure solvent. 

Osmotic term (Ilion) neglecting of charge-charge interaction as well as counterion condensation Debye-Hiickel model (Hasa-Ilavsky-Dusek theory) Dusek et al. [44]... 

Do activities and solubilities calculated from this alternate approach differ from those calculated with association-equilibrium extended-Debye-Hiickel models under conditions of interest in FGD wet scrubbing ... 

In the simple Debye-Hiickel model, attention is focused on the coulombic interactions as the source of non-ideality. The aim of the Debye-Hiickel theory is to calculate the mean activity coefficient for an electrolyte in terms of these electrostatic interactions between the... 

These points are the main features of the simple Debye-Hiickel model. Other aspects of the Debye-Hiickel theory are illustrated by the mode of approach considered shortly. 

Meanwhile, it is constmctive to look again at the physical basis of the simple Debye-Hiickel model and its mathematical development to see where both could be modified, and to consider whether this would be mathematically possible. What has been written in Chapter 1 on ions and solvent structure shows that the Debye-Huckel model is painfiiUy naive and cannot even approach physical reality. A brief reassessment of the features 1-7 of the simple Debye-Hiickel model is given below, along with indications as to how these problems have been tackled. 

Bjerrum looks at the direct interaction between an ion of charge zie and another ion with charge z-ie leading to the formation of the ion pair. The calculation is based on the Debye-Hiickel model (Section 10.3) and its development as given in Section 10.6. 

If all sorts of possible combinations of the features suggested above are to be incorporated in the calculations and each result is compared with the others and with experiment, a much deeper and more accurate picture of what is happening at the microscopic level emerges. Consequently, a better theory of electrolyte solutions has been forthcoming. Much highly promising work has been done and modifications both to the Debye-Hiickel model and its... 

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