Debye–Hückel theory

The Debye–Hückel theory was proposed by Peter Debye and Erich Hückel as a theoretical explanation for departures from ideality in solutions of electrolytes and plasmas.

In general, the mean activity coefficient of a fully dissociated electrolyte of formula AnBm is given by[4]

A description of Debye–Hückel theory includes a very detailed discussion of the assumptions and their limitations as well as the mathematical development and applications.

It is also assumed that The last assumption means that each cation is surrounded by a spherically symmetric cloud of other ions.

[7] The deviation from ideality is taken to be a function of the potential energy resulting from the electrostatic interactions between ions and their surrounding clouds.

This distribution also depends on the potential ψ(r) and this introduces a serious difficulty in terms of the superposition principle.

The multiple-charge generalization from electrostatics gives an expression for the potential energy of the entire solution.

is expressed in terms of molality, instead of molarity (as in the equation above and in the rest of this article), then an experimental value for

The most significant aspect of this result is the prediction that the mean activity coefficient is a function of ionic strength rather than the electrolyte concentration.

In this situation the mean activity coefficient is proportional to the square root of the ionic strength.

gives satisfactory agreement with experimental measurements for low electrolyte concentrations, typically less than 10−3 mol/L.

They usually allow the Debye–Hückel equation to be followed at low concentration and add further terms in some power of the ionic strength to fit experimental observations.

The Debye–Hückel equation cannot be used in the solutions of surfactants where the presence of micelles influences on the electrochemical properties of the system (even rough judgement overestimates γ for ~50%).

When conductivity is measured the system is subject to an oscillating external field due to the application of an AC voltage to electrodes immersed in the solution.

In addition it was assumed that the electric field causes the charge cloud to be distorted away from spherical symmetry.

An English translation[21]: 217–63  of the article is included in a book of collected papers presented to Debye by "his pupils, friends, and the publishers on the occasion of his seventieth birthday on March 24, 1954".

In the following summary (as yet incomplete and unchecked), modern notation and terminology are used, from both chemistry and mathematics, in order to prevent confusion.

D&H note that the Guldberg–Waage formula for electrolyte species in chemical reaction equilibrium in classical form is[21]: 221

may be derived from the functional dependence of the chemical potential of a component of an ideal mixture upon its mole fraction.

of a solution is lowered by the electrical interaction of its ions, but that this effect can't be determined by using the crystallographic approximation for distances between dissimilar atoms (the cube root of the ratio of total volume to the number of particles in the volume).

Note that in the infinite temperature limit, all ions are distributed uniformly, with no regard for their electrostatic interactions.

This equation is difficult to solve and does not follow the principle of linear superposition for the relationship between the number of charges and the strength of the potential field.

It has been solved analyticallt by the Swedish mathematician Thomas Hakon Gronwall and his collaborators physical chemists V. K. La Mer and Karl Sandved in a 1928 article from Physikalische Zeitschrift dealing with extensions to Debye–Huckel theory.

[25] However, for sufficiently low concentrations of ions, a first-order Taylor series expansion approximation for the exponential function may be used (

Lastly, they claim without proof that the addition of more terms in the expansion has little effect on the final solution.

D&H recognize the importance of the parameter in their article and characterize it as a measure of the thickness of the ion atmosphere, which is an electrical double layer of the Gouy–Chapman type.

However, one can see that this is the case by considering that any spherical static charge distribution is subject to the mathematics of the shell theorem.

[28] Since the ion atmosphere is assumed to be (time-averaged) spherically symmetric, with charge varying as a function of radius

To verify the validity of the Debye–Hückel theory, many experimental ways have been tried, measuring the activity coefficients: the problem is that we need to go towards very high dilutions.

Going towards high dilutions good results have been found using liquid membrane cells, it has been possible to investigate aqueous media 10−4 M and it has been found that for 1:1 electrolytes (as NaCl or KCl) the Debye–Hückel equation is totally correct, but for 2:2 or 3:2 electrolytes it is possible to find negative deviation from the Debye–Hückel limit law: this strange behavior can be observed only in the very dilute area, and in more concentrate regions the deviation becomes positive.