## Tuesday, January 22, 2008

### Debye-Hückel Theory

One of the problems that arises when treating the theory of ionic solutions is how exactly to deal with the activities of cations versus anions when both are always found together (all solutions are neutral). Recognizing this fact, we can define a mean activity coefficient in terms of the individual ion activity coefficients:

γ± = [γ+ν+γ+ν-]1/ν

But how do we obtain activity coefficients without going through tedious cryoscopic or electrochemical measurements? This is one of the results that the Debye-Hückel theory seeks to uncover. Before going through the details, we need a convenient way to describe how much a given solution can conduct electricity: the ionic strength I.

I = 1/2 Σ zi2 (mi/m°) = 1/2 Σ zi2 mi

Ionic strength is unitless so the last term above is basically a unitless molality.

Debye and Hückel proposed a simple model that they hoped would capture most of the behavior of ionic solutions. The assumptions:
1. an ionic atmosphere is established with many ions with valence z and radius a
2. the solvent is structureless with electric permittivity ε
3. the solution is dilute.
Several results were obtained via mind-bleeding statistical mechanics but among them were the Debye length τD, which is a measure of the effective range that ionic charges interact in this solution.

Another important result is the extended Debye-Hückel law:

lnγ± = - |z+z-|αI1/2/(1 + βaI1/2)

which can also be written in ion activity form:

lnγi = -zi2|αI1/2/(1 + βaI1/2)

As mentioned in class, this formula works quite well for dilute to moderately concentrated solutions (and is fairly bad past 0.25 m). For dilute solutions we can consider the limiting Debye-Hückel law:

lnγ± = - |z+z-|αI1/2

Since α=1.171 for water at 25°C, we can transmogrify any of these formulas to log10 world. For example, the above equation becomes

logγ± = -0.509 |z+z-|I1/2 (aqueous solutions, 25°C)

Bjerrum, Mayer and Davies extended the Debye-Hückel formula so that it works for a wide range of ionic strengths (and remains the primary equation used in pchem today):

logγ± = -0.509 |z+z-|(I1/2/1+I1/2 - 0.30I) (aqueous solutions, 25°C)

At different temperatures, new values for the Debye-Hückel parameter (the 0.509 value) would need to be calculated through α.

Tomorrow, salting-in and salting-out.