The Debye Huckel theory shows how the potential energy of an ion in solution depends on the ionic strength of the solution.
Except at infinite dilution, electrostatic interaction between ions alters the properties of the solution from those excepted from the free ion model that leads to ideal behavior.
The treatment of ion-ion interactions by Debye-Huckel in 1923 and 1924 led to an explanation of the properties of relatively dilute solutions, less than about 0.01 M. even this limited success has provided valuable in such a way to expolarate available experimental data to the limit of infinite dilution was provided. Also provided was a base for one empirical extension to higher concentrations. It is worthwhile, therefore, to follow through the Debye-Huckel derivation in some detail.
The Debye Huckel treatment deals with the distribution of ions around a given ion and the net effect these neighboring ions have on the ions of the solution.
Consider one of the ions, a positive ion to be specific, of an aqueous solution of an electrolyte. It will be affected by coulombic interactions with the other ions of the solution. These interactions are described by a potential energy term that is distance between them. Thus a nearby ion will have a greater effect on the reference ion than a far off ion will. But the number of such distant ions increases as the volume of a spherical shell, i.e. as r2. Thus these more distant ions, and therefore the bulk of the solution, might appear to require our attention in the deduction of the affect on the reference ion. Fortunately, ions of opposite charge can be expected to distribute themselves uniformly at some distance from a given ion to produce electrical neutrality well removed from a given ion.
Consider first how the ions in a solution distribute themselves relative to each other. The two factors that determine the distribution are thermal jostlings and the electric interaction between charged particles. Suppose that on the average there are n, ions of the I type per unit volume. Around any positive ion there will be an increase in the concentration of negative ions and a decrease in the concentration of positive ions. These changes results from the ions moving to the energetically more favored regions, i.e. those in which their potential energy is low. The tendency for this movement must complete with the traditional thermal motion.
Boltzmann's equation can be used to give the number of ions that on average are a distance r from the positive charge. The energy of ions of charge Zie in a potential value is (eZi) potential value. If Zi is positive, the energy is higher near the reference positive charge; if Zi is positive, the energy is lower. Boltzmann's equation gives:
Ni (r)= nie -(eZi∫)/ (kT)
Where ni (r) is the number of ith ions per unit volume at a distance r from the reference positive charge and ni is the average number per unit volume in the solution.
This expression cannot be used directly to calculate the density of ions of each type in the neighborhood of the reference ion, determine the potential ∫. Some manipulation of expression for change densities and potentials is necessary to get around this difficulty.
To provide a reference for this more complex situation, let us first describe the density of ions about the reference positive ion if only the reference change affected the distribution. Then we could write:
∫ = e/(4∏ε)/r
Where e is the charge of the reference ion. The number of ions of type I per unit volume at a distance r form the reference ion is now given by;
ni (r) = nie -e2Zi/(4∏εrkT)
or,
ni (r) /ni = e -e2Zi/(4∏εrkT)
the results of the above eq. must be corrected to take into account the effect of the ions that surround the reference ion. To do so, we develop an expression for the net excess negative charge in the region around the reference, positively charged ion.
Charge density around the reference ion: the charge density at a distance r from the unit positive charge given by the number of ions ni(r) of a particular type times the charge eZi of these ions. Thus the charge density can be expressed by:
p(r) = Σ (eZi)ni(r)
if the potential at a distance r is ∫ eq, can be used to write:
p(r) = e Σi ni Zie -eZi∫/(kT)