Debye-Hiickel Theory of Ionic Solutions

Ionic strength is a useful concept because it allows us to consider some general expressions that depend only on ionic strength and not on the identities of the ions themselves. In 1923, Peter Debye and Erich Hiickel made some simplifying assumptions about all ionic solutions. They assumed that they would be dealing with very dilute solutions, and that the solvent was basically a continuous, structureless medium that has some dielectric constant e. Debye and Hiickel also assumed that any deviations in solution properties from ideality were due to the coulombic interactions (repulsions and attractions) between the ions. 

Applying some of the tools of statistics and the concept of ionic strength, Debye and Hiickel derived a relatively simple relationship between the activity coefficient y+ and the ionic strength / of a dilute solution  

Equation 8.50 is the central part of what is called the Debye-Hiickel theory of ionic solutions. Because it strictly applies only to dilute solutions (/ 0.01m), this expression is more specifically known as the Debye-Hiickel limiting law. Because A is always positive, the product of the charges 2+ is always negative, so In y+ is always negative. This implies that y+ is always less than 1, which in turn implies that the solution is not ideal. 

Unless otherwise noted, all art on this page is Cengage Learning 2014 

FIG U RE 8.9 Plot of experimental values of ln(y+) versus 7 2 in aqueous solution. The solid black lines show the predictions of the Debye-Huckel limiting law, while the colored lines show the experimental values for three different salts. Note how the Debye-Hiickel limiting law holds only for dilute solutions. 

---

Improvements upon the Debye-Hiickel Theory of Ionic Solutions Andersen, H. C. 11... 

Debye length - In the Debye-Hiickel theory of ionic solutions, the effective thickness of the cloud of ions of opposite charge which surrounds each given ion and shields the Coulomb potential produced by that ion. 

The first-order term involves the pair potential of average force. This may be sometimes approximated by viewing the solvent as a continuum. An example of such an approximation is the Debye-Hiickel theory of ionic solutions. 

Evidently there are factors at work in an electrolytic solution that have not yet been reckoned with, and the ion size parameter is being asked to include the effects of all these factors simultaneously, even though these other factors probably have little to do with the size of the ions and may vary with concentration. If this were so, the ion size parameter a, calculated back from experiment, would indeed have to vary with concentration. The problem therefore is What factors, forces, and interactions were neglected in the Debye-Hiickel theory of ionic clouds ... 

Debye Length A parameter in the Debye—Hiickel theory of electrolyte solutions, k-1. For aqueous solutions at 25 °C, k = 3.288y7 in reciprocal nanometers, where I is the ionic strength of the solution. The Debye length is also used in the DLVO theory, where it is referred to as the electric double-layer thickness. See also Electric Double-Layer Thickness. 

Stress has been laid on the contribution of Debye and Hiickel (1923) to the development of the theory of ion-ion interactions. It was Debye and Hiickel who ushered in the electrostatic theory of ionic solutions and worked out predictions that precisely fitted experiments for sufficiently low concentrations of ions. It is not often realized, however, that the credit due to Debye and Hiickel as the parents of the theory of ionic solutions is the credit that is quite justifiably accorded to foster parents. The true parents were Milner and Gouy. These authors made important contributions very early in the growth of the theory of ion-ion interactions. 

The work of Debye and Erich Hiickel (1896-1880), published in 1923, led to a theory of ionic solutions that explained a number of anomalies concerning conductivities of electrolytic solutions. In 1926, Lars Onsager (1903-76) added the treatment of Brownian motion toward understanding the transport properties of ions in melts, aqueous, and... 

The concept behind the DH theory was not new, in that Milner (3a), almost a decade before, formulated a theory of ionic solutions based on the concept of "ionic atmosphere". He, however, was unable to solve the proposed equations. Double layer theories(3b,3c), which used the same concept, also preceded the DH theory. The merit of Debye and Hiickel was to introduce several approximations that made an analytical solution for the theory possible. The starting point of the DH theory is the assumption that the excess of thermodynamic properties of electrolyte solutions (when compared with non-electrolyte solutions) is due only to the Coulombic interactions between the ions. It is then necessary to calculate the average electrostatic potential at the surface of a given ion (taken as reference) due to all the other ions. These other ions constitute the "ionic atmosphere". Once this potential is known, it is evidently possible to calculate all the thermodynamic properties of the system. Indicating with z e and zje the charge of the reference ion (i) and of an arbitrary ion (j) in the "ionic atmosphere", respectively, the effective interaction energy between the two ions will be... 

A theory of ionic solutions developed by Peter Debye and Erich Hiickel in 1923 (which is based on statistical mechanics and beyond the scope of this text) provides an expression for the activity. We shall only state the main result of this theory, which works well for dilute electrolytes. The activity depends on a quantity called the ionic strength /, defined by... 

Guoy and Chapman developed a theory of the charge distribution in the double layer about 10 years before Debye and Hiickel developed their theory of ionic solutions, which is quite similar to it. If one neglects nonelectrostatic contributions to the potential energy of an ion of type i with valence Zi, the concentration of ions of type i in a region of electric potential cp is given by the Boltzmann probability formula, Eq. (9.3-41) ... 

In Fig. 69 we have been considering a pair of solute particles in pure solvent. We shall postpone further discussion of this question until later. In the meantime we shall review the Coulomb forces in very dilute ionic solutions as they are treated in the Debye-Hiickel theory. 

Experience shows that solutions of other electrolytes behave in a manner similar to the examples we have used. The conclusion we reach is that the Debye-Hiickel equation, even in the extended form, can be applied only at very low concentrations, especially for multivalent electrolytes. However, the behavior of the Debye-Hiickel equation as we approach the limit of zero ionic strength appears to give the correct limiting law behavior. As we have said earlier, one of the most useful applications of Debye-Hiickel theory is to... 

Equation (7.44) is known as the third approximation of the Debye-Hiickel theory. Numerous attempts have been made to interpret it theoretically, hi these attempts, either individual simplifying assumptions that had been made in deriving the equations are dropped or additional factors are included. The inclusion of ionic solvation proved to be the most important point. In concentrated solutions, solvation leads to binding of a significant fraction of the solvent molecules. Hence, certain parameters may change when solvation is taken into account since solvation diminishes the number of free solvent molecules (not bonded to the ions). The influence of these and some other factors was analyzed in 1948 by Robert A. Robinson and Robert H. Stokes. 

Another arena for the application of stochastic frictional approaches is the influence of ionic atmosphere relaxation on the rates of reactions in electrolyte solutions [19], To gain perspective on this, we first recall the early and often quoted triumph of TST for the prediction of salt effects, in connection with Debye-Hiickel theory, for reaction rates In kTST varies linearly with the square root of the solution ionic strength I, with a sign depending on whether the charge distribution of the transition state is stabilized or destabilized by the ionic atmosphere compared to the reactants. 

Experimenters would do well to avoid any unnecessary changes in the ionic composition of reaction samples within a series of experiments. If possible, chose a standard set of reaction conditions, because one cannot readily correct data from one set of experimental conditions in any reliable manner that reveals the reactivity under a different set of conditions. Maintenance of ionic strength and solvent composition is desirable, and correction to constant ionic strength often effectively minimizes or ehminates electrostatic effects. Even so, remember that Debye-Hiickel theory only applies to reasonably dilute electrolyte solutions. Another important fact is that ion effects and solvent effects on the activity coefficients of polar transition states may be more significant than more modest effects on reactants. 

Recall from transition state theory that the rate of a reaction depends on kg (the catalytic rate constant at infinite dilution in the given solvent), the activity of the reactants, and the activity of the activated complex. If one or more of the reactants is a charged species, then the activity coefficient of any ion can be expressed in terms of the Debye-Htickel theory. The latter treats the behavior of dilute solutions of ions in terms of electrical charge, the distance of closest approach of another ion, ionic strength, absolute temperature, as well as other constants that are characteristic of each solvent. If any other factor alters the effect of ionic strength on reaction rates, then one must look beyond Debye-Hiickel theory for an appropriate treatment. 

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. 

Normally, the validity of the Debye—Hiickel theory extends little further than kR <1. At room temperature, this requires ionic concentrations < 0.1 mol dm-3 for univalent ions in water, 0.03moldm-3 for univalent ions in ethanol or <0.01 mol dm-3 for univalent ion in ethers. In these cases, ions may be regarded as point particles and the strong repulsive core potential ignored. Furthermore, the time taken for non-reactive ions to diffuse far enough to establish an ionic-atmosphere around an ion, which was suddenly formed in solution containing only univalent ions, is... 

In 1923, Peter Debye and Erich Hiickel developed a classical electrostatic theory of ionic distributions in dilute electrolyte solutions [P. Debye and E. Hiickel. Phys. Z 24, 185 (1923)] that seems to account satisfactorily for the qualitative low-ra nonideality shown in Fig. 8.3. Although this theory involves some background in statistical mechanics and electrostatics that is not assumed elsewhere in this book, we briefly sketch the physical assumptions and mathematical techniques leading to the Debye-Hiickel equation (8.69) to illustrate such molecular-level description of thermodynamic relationships. 

---