Debye and Hiickel

Debye-Hiickel theory The activity coefficient of an electrolyte depends markedly upon concentration. Jn dilute solutions, due to the Coulombic forces of attraction and repulsion, the ions tend to surround themselves with an atmosphere of oppositely charged ions. Debye and Hiickel showed that it was possible to explain the abnormal activity coefficients at least for very dilute solutions of electrolytes. 

The method attributed to Debye and Hiickel has been almost universally adopted by scientists. We will review the steps of their method and give the equations that are the end product of the derivation, but will leave it to others to present the mathematical details.6... 

In the second approximation, Debye and Hiickel introduced the idea that the centers of the ions cannot come closer than a certain minimum distance a, which depends on ion size the ions were now treated as entities with a finite radius. The mathematical result of this assumption are charge densities Qy, which are zero for r[Pg.120]

It may seem that the prospeets are bleak for the GvdW approach to electrolytes but, in fact, the reverse is the ease. We need only follow Debye and Hiickel [18] into their analysis of the sereening meehanism, almost as successful as the van der Waals analysis of short-range fluids, to see that the mean-field approximation can be applied to the correlation mechanism with great advantage. In fact, we can then add finite ion size effects to the analysis and thereby unify these two most successful traditional theories. 

The theory of Debye and Hiickel started from the assumption that strong electrolytes are completely dissociated into ions, which results, however, in electrical interactions between the ions in such a manner that a given ion is surrounded by a spherically symmetrical distribution of other ions mainly of opposite charges, the ionic atmosphere. The nearer to the central ions the higher will be the potential U and the charge density the limit of approach to the central ion is its radius r = a. 

J. Bjerrum (1926) first developed the theory of ion association. He introduced the concept of a certain critical distance between the cation and the anion at which the electrostatic attractive force is balanced by the mean force corresponding to thermal motion. The energy of the ion is at a minimum at this distance. The method of calculation is analogous to that of Debye and Hiickel in the theory of activity coefficients (see Section 1.3.1). The probability Pt dr has to be found for the ith ion species to be present in a volume element in the shape of a spherical shell with thickness dr at a sufficiently small distance r from the central ion (index k). 

At the final concentration c, the potential j)k at distance r is given not only by the potential of the ion, )°ky but also by the potentials of the surrounding ions. Debye and Hiickel assumed that a spherical ionic atmosphere of statistically prevailing ions with opposite charge forms around each ion, giving rise to the potential ipa. Thus xpk = ipk + The potential of the space charge density p is given by the Poisson equation... 

If the origin of the coordinate system is located in the centre of the central kth ion, the number of particles dNt in the volume dV is given by a distribution function, expressed by Debye and Hiickel in terms of the Boltzmann distribution law (cf. p. 215)... 

The first approximate calculation was carried out by Debye and Hiickel and later by Onsager, who obtained the following relationship for the relative strength of the relaxation field AE/E in a very dilute solution of a single uni-univalent electrolyte... 

In dilute solutions it is possible to relate the activity coefficients of ionic species to the composition of the solution, its dielectric properties, the temperature, and certain fundamental constants. Theoretical approaches to the development of such relations trace their origins to the classic papers by Debye and Hiickel (6-8). For detailpd treatments of this subject, refer to standard physical chemistry texts or to treatises on electrolyte solutions [e.g., that by Harned... 

When electrical attraction and repulsion operate over distances considerably larger than the hydrated sizes of the ions, we can compute species activities quite well from electrostatic theory, as demonstrated in the 1920s by the celebrated physical chemists Debye and Hiickel. At moderate concentrations, however, the ions pack together rather tightly. In a one molal solution, for example, just a few... 

In 1923, Debye and Hiickel published their famous papers describing a method for calculating activity coefficients in electrolyte solutions. They assumed that ions behave as spheres with charges located at their center points. The ions interact with each other by coulombic forces. Robinson and Stokes (1968) present their derivation, and the papers are available (Interscience Publishers, 1954) in English translation. 

While several investigators contributed substantially to the resolution of this problem, it was the classic work of Debye and Hiickel ( 1) which provided a simple yet adequate explanation of the effect on thermodynamic properties of the long-range electrostatic forces between ions in solution. The experimental work of that era tended to emphasize dilute solutions at room temperature. 

The theory of complete electrolytic dissociation at infinite dilution was developed by Debye and Hiickel (1923) and has been further extended by Onsager and later by Fuoss and Krauss. 

Electrostatic and statistical mechanics theories were used by Debye and Hiickel to deduce an expression for the mean ionic activity (and osmotic) coefficient of a dilute electrolyte solution. Empirical extensions have subsequently been applied to the Debye-Huckel approximation so that the expression remains approximately valid up to molal concentrations of 0.5 m (actually, to ionic strengths of about 0.5 mol L ). The expression that is often used for a solution of a single aqueous 1 1, 2 1, or 1 2 electrolyte is... 

Debye and Hiickel (1923) used a similar approach to Gouy and Chapman to calculate the activity coefficients of electrolytes. 

The theory of Debye and Hiickel has survived much criticism since the appearance of their celebrated paper (I). This is no doubt because of the simplicity and essential correctness of the limiting laws (2,3,4). Nevertheless, many modifications of their treatment have failed to provide a convincing picture of the interionic effects and structure in the concentration range of practical importance (5, 6). The work presented here was stimulated by the difficulties of extrapolation encountered in a mixed-solvent emf study (7), and contradicts current trends suggesting that the inadequacy of the DH theory for all but very dilute solutions springs solely from the crudity of the original model. The authors propose a more realistic model that allows the ions to be polarizable and leads to markedly different results. 

Figure la. Left a segment of the spherical ionic atmosphere. Right the reduced ionic cloud of Debye and Hiickel. 

The following calculations are based on the Debye-Hiickel (1) expression for the field of an ion and on the corresponding expression for the field of a dipole as formulated by the authors (2). However, it will be necessary to modify the field of Debye and Hiickel for the interior of the ion to make it conform to the model used by the authors. 

If the ions are idealized as spheres of radius aj, the field of Debye and Hiickel is given by... 

In the original treatment of Debye and Hiickel these constants were determined under the assumption that the ion had a point charge at r = 0 and that the interior of the ion had the same dielectric constant D as the solvent. In the On-sager (5) theory of dipolar liquids it is assumed that the molecule is represented by a spherical cavity in the liquid with a singularity at its center. The characteristics of the molecule are its electric moment in vacuum po and its polarizability a. This is to be related to an internal refractive index n by... 

So far, very dilute solutions have been considered such that the interaction between ions is only coulombic. When other (unreactive) ions are nearby, the direct interaction between ionic reactants is partially screened and was first developed by Debye and Hiickel [91]. They showed that the potential energy, eqn. (39), is modified and becomes... 

In the years 1910-1917 Gouy2 and Chapman3 went a step further. They took into account a thermal motion of the ions. Thermal fluctuations tend to drive the counterions away form the surface. They lead to the formation of a diffuse layer, which is more extended than a molecular layer. For the simple case of a planar, negatively charged plane this is illustrated in Fig. 4.1. Gouy and Chapman applied their theory on the electric double layer to planar surfaces [54-56], Later, Debye and Hiickel calculated the potential and ion distribution around spherical surfaces [57],... 

This simple equation is, however, only valid for R Xp- If the radius is not much larger than the Debye length we can no longer treat the particle surface as an almost planar surface. In fact, we can no longer use the Gouy-Chapman theory but have to apply the theory of Debye and Hiickel. Debye and Hiickel explicitly considered the electric double layer of a sphere. A result of their theory is that the total surface charge and surface potential are related by... 

K+ + SOJ2) however much Debye and Hiickel use ion activities in their theory of long-distance interactions in dielectric solvents. Niels Bjerrum172) wrote a paper about this paradox which extends very far a well-known concept such as pH is not defined, strictly speaking, without reference to the surrounding salt medium and in particular to the ambient anions. 

This is the screened potential originally developed by Debye and Hiickel [119]. 

The three terms in these equations reading from left to right are related to 7, a , and to of Eq. 2.13, respectively. The activity coefficient and the osmotic coefficient measure the degree to which solute concentrations and the activity of water (aw) depart from ideal solutions, respectively. For ideal solutions, a = to and 7 = 1.0 (Eq. 2.13) or Gex = 0 (Eq. 2.32). Similarly, aw = 1.0 for an ideal solution. In the real world, solutions are rarely ideal, except in the infinitely dilute case we therefore need a model for calculating and (f>[= f(aw)]. An early model based on statistical mechanics was developed by Debye and Hiickel (1923). Their equations are... 

C) which he derived from the ionic product of water (Kw = 10 14 mol x dm 3). Some years later, Lewis introduced the concept of activity, and in 1923 Debye and Hiickel published their theory for strong electrolyte solutions. On the basis of this knowledge, Soerensen and Linderstroem-Lang [2] suggested a new pH definition in terms of the relative activity of hydrogen ions in solution ... 

With increasing ionic strength, some charges of the protein particle are neutralized by buffer ions of the opposite sign, according to the theory of Debye and Hiickel, and mobility consequently decreases. Mathematical expression of this phenomenon may be given by the equation of Audubert (A8) ... 

Pulsating DC at 80 kilocycles was used by Mach and Geffert (Ml), and it shortened the duration of the run by about one-third, perhaps as the result of the disruption of the Debye and Hiickel ion cloud. This so-called Wien effect has not yet been found by other authors (D17). 

According to the interionic attraction theory of Debye and Hiickel... 

In all other solutions the so called degree of dissociation, as determined from the measurement of some colligative property, merely indicates the magnitude of interionic forces, it cannot, however, be taken as a measure of the quantity of dissociated and undissociated molecules of the solute. A complete theory of strong electrolytes, at least of their diluted solutions, has been developed by Debye and Hiickel, this theory is the basis of modern electrochemistry. 

Debye and Hiickel were working out their conceptions mathematically, and found that the mean activity coefficient of a strong electrolyte A y B 7 may be expressed by the equation ... 

Although Debye and Hiickel worked out their theory to solve the problem of strong, completely dissociated electrolytes, the results may be applied to weak and transition electrolytes as well, if the actual ionic concentration is substituted in the equation for ionic strength. With strong electrolytes, which are completely dissociated, it is possible to substitute in the term directly the analytical concentration of the substance, but with weak electrolytes their dissociation degree a has to be considered. For example with uni-... 

Combining (29) and (30) we have Debye and Hiickel s relation between velocity and potential... 

Debye-Huckel approximation — In calculating the potential distribution around a charge in a solution of a strong -> electrolyte, - Debye and -> Hiickel made the assumption that the electrical energy is small compared to the thermal energy ( zjei (kT), and they solved the -> Poisson-Boltzmann equation V2f = - jT- gc° eexp( y) by expanding the exponential... 

Debye-Hiickel theory — The interactions between the ions inside an electrolyte solution result in a nonideal behavior as described with the concepts of mixed-phase thermodynamics. Assuming only electrostatic (i.e., coulombic) interactions - Debye and - Hiickel suggested a model describing these interactions resulting in - activity coefficients y suitable for further thermodynamic considerations. Their model is based on several simplifications ... 

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