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The chemists Peter Debye and Erich Hückel noticed that solutions that contain ionic solutes do not behave ideally even at very low concentrations. So, while the concentration of the solutes is fundamental to the calculation of the dynamics of a solution, they theorized that an extra factor that they termed gamma is necessary to the calculation of the activity coefficients of the solution. Hence they developed the Debye–Hückel equation and Debye–Hückel limiting law. The activity is only proportional to the concentration and is altered by a factor known as the activity coefficient
Contents
Debye–Hückel limiting law
For the principles used to derive this equation see Debye–Hückel theoryIn order to calculate the activity,
where
The Debye–Hückel limiting law enables one to determine the activity coefficient of an ion in a dilute solution of known ionic strength. The equation is[1]
It is important to note that because the ions in the solution act together, the activity coefficient obtained from this equation is actually a mean activity coefficient.
The excess osmotic pressure which is obtained from Debye–Hückel theory is in cgs units:
Therefore the total pressure is the sum of the excess osmotic pressure and the ideal pressure
Summary of Debye and Hückel's first paper on the theory of dilute electrolytes
The English title of the paper is called "On the Theory of Electrolytes. I. Freezing Point Depression and Related Phenomena." It was originally published in volume 24 of a German-language journal, called Physikalische Zeitschrift, in 1923. An English translationof the paper 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." The paper deals with the calculation of properties of electrolyte solutions that are not under the influence of net electric fields, thus it deals with electrostatics.
In the same year they first published this paper, Debye and Hückel, hereinafter D&H, also released a paper that covered their initial characterization of solutions under the influence of electric fields called "On the Theory of Electrolytes. II. Limiting Law for Electric Conductivity," but that subsequent paper is not (yet) covered here.
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. Also, with a few exceptions to improve clarity, the subsections in this summary are (very) condensed versions of the same subsections of the original paper.
Introduction
D&H note that the Guldberg–Waage formula for electrolyte species in chemical reaction equilibrium in classical form is
D&H say that, due to the "mutual electrostatic forces between the ions," it is necessary to modify the Guldberg–Waage equation by replacing
The relationship between
Fundamentals
D&H use the Helmholtz and Gibbs free entropies,
D&H give the total differential of
By the definition of the total differential, this means that
which are useful further on.
As stated previously, the internal energy is divided into two parts,
Similarly, the Helmholtz free entropy is also divided into two parts,
D&H state, without giving the logic, that
It would seem that, without some justification,
Without mentioning it specifically, D&H later give what might be the required (above) justification while arguing that
The definition of the Gibbs free entropy,
D&H give the total differential of
At this point D&H note that, for water containing 1 mole per liter of potassium chloride (nominal pressure and temperature aren't given), the electric pressure,
and put
D&H say that, according to Planck, the classical part of the Gibbs free entropy is
Species zero is the solvent. The definition of
D&H don't say so, but the functional form for
D&H note that the internal energy,
The potential energy of an arbitrary ion solution
Electroneutrality of a solution requires that
To bring an ion of species i, initially far away, to a point
at every point in the cloud. Note that in the infinite temperature limit, all ions are distributed uniformly, with no regard for their electrostatic interactions.
The charge density is related to the number density:
When combining this result for the charge density with the Poisson equation from electrostatics, a form of the Poisson–Boltzmann equation results:
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 by the Swedish mathematician Thomas Hakon Gronwall and his collaborators physicical chemists V.K. La Mer and Karl Sandved in a 1928 paper from Physikalische Zeitschrift which deals with extensions to Debye-Huckel theory which resorted to Taylor series expansion.
However, for sufficiently low concentrations of ions, a first order Taylor series expansion approximation for the exponential function may be used (
The Poisson–Boltzmann equation is transformed to
because the first summation is zero due to electroneutrality.
Factor out the scalar potential and assign the leftovers, which are constant, to
So, the fundamental equation is reduced to a form of the Helmholtz equation:(http://guava.physics.uiuc.edu/~nigel/courses/569/Essays_2004/files/lu.pdf section 3.1)
Today,
The equation may be expressed in spherical coordinates by taking
The equation has the following general solution; keep in mind that
The electric potential is zero at infinity by definition, so
In the next step, D&H assume that there is a certain radius,
The potential of a point charge by itself is:
D&H say that the total potential inside the sphere is
where
In a combination of the continuously distributed model which gave the Poisson–Boltzmann equation and the model of the point charge, it is assumed that at the radius
By the definition of electric potential energy, the potential energy associated with the singled out ion in the ion atmosphere is
Notice that this only requires knowledge of the charge of the singled out ion and the potential of all the other ions.
To calculate the potential energy of the entire electrolyte solution, one must use the multiple charge generalization for electric potential energy.
Nondimensionalization
This section was created without reference to the original paper and there are some errors in it (for instance, the ionic strength is off by a factor of two). Once these are rectified, this section should probably be moved to the nondimensionalization article and then be linked from here, since the nondimensional version of the Poisson–Boltzmann equation isn't necessary to understand the D&H theory.
The differential equation is ready for solution (as stated above, the equation only holds for low concentrations):
Using the Buckingham π theorem on this problem results in the following dimensionless groups:
To obtain the nondimensionalized differential equation and initial conditions, use the
For table salt in 0.01 M solution at 25 °C, a typical value of
Experimental verification of the theory
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. Typical examples are: measurements of vapour pressure, freezing point, osmotic pressure (indirect methods) and measurement of electric potential in cells (direct method). 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. It is possible that Debye–Hückel equation is not able to foresee this behavior because of the linearization of the Poisson–Boltzmann equation, or maybe not: studies about this have been started only during the last years of the 20th century because before it wasn’t possible to investigate the 10−4 M region, so it is possible that during the next years new theories will be born.
Extensions of the theory
Warning: The notation in this section is (currently) different from in the rest of the article.
A number of approaches have been proposed to extend the validity of the law to concentration ranges as commonly encountered in chemistry
One such extended Debye–Hückel equation is given by:
where
The extended Debye–Hückel equation provides accurate results for μ ≤ 0.1. For solutions of greater ionic strengths, the Pitzer equations should be used. In these solutions the activity coefficient may actually increase with ionic strength.
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%).