Thermodynamics and Chemistry

CHAPTER 10 ELECTROLYTE SOLUTIONS

10.5 DERIVATION OF THEDEBYE–HUCKEL ̈ EQUATION 297

The ionic strengthImis calculated from Eq.10.4.2with the molalities ofallions in the
solution, not just the molality of the ion or solute whose activity coefficient we are interested

in. This is because, as explained above, the departure of (^) Cand (^) from the ideal-dilute
value of 1 is caused by the interaction of each ion with the ion atmosphere resulting from
all other ions in the solution.
In a binary solution of a single electrolyte solute, assumed to be completely dissociated,
the relation between the ionic strength and the solute molality depends on(the number of
ions per solute formula unit) and the charge numberszCandz. The ionic strength is given
byImD.1=2/

P

imiz

2

iD.1=2/.Cz

2

CCz

2
/mB. With the help of the electroneutrality
conditionCzCD.z/, this becomes

ImD^12 å.z/zC.CzC/zçmB

D^12 å.CC/zCzçmB

D^12 

(^) zCz
(^) mB (10.4.9)
We find the following relations hold betweenImandmBin the binary solution, depending
on the stoichiometry of the solute formula unit:
For a 1:1 electrolyte, e.g., NaCl or HCl:ImDmB
For a 1:2 or 2:1 electrolyte, e.g., Na 2 SO 4 or CaCl 2 :ImD3mB
For a 2:2 electrolyte, e.g., MgSO 4 :ImD4mB
For a 1:3 or 3:1 electrolyte, e.g., AlCl 3 :ImD6mB
For a 3:2 or 2:3 electrolyte, e.g., Al 2 (SO 4 ) 3 :ImD15mB
Figure10.4on the next page shows ln (^) as a function of
p
Imfor aqueous HCl and
CaCl 2. The experimental curves have the limiting slopes predicted by the Debye–Huckel ̈
limiting law (Eq.10.4.8), but at a low ionic strength the curves begin to deviate significantly
from the linear relations predicted by that law. The full Debye–Huckel equation (Eq. ̈ 10.4.7)
fits the experimental curves over a wider range of ionic strength.

10.5 Derivation of the Debye–Huckel Equation ̈

Debye and H ̈uckel derived Eq.10.4.1using a combination of electrostatic theory, statis-
tical mechanical theory, and thermodynamics. This section gives a brief outline of their
derivation.
The derivation starts by focusing on an individual ion of speciesias it moves through
the solution; call it the central ion. Around this central ion, the time-average spatial dis-
tribution of any ion speciesjis not random, on account of the interaction of these ions of
speciesjwith the central ion. (Speciesiandjmay be the same or different.) The distribu-
tion, whatever it is, must be spherically symmetric about the central ion; that is, a function
only of the distancerfrom the center of the ion. The local concentration,cj^0 , of the ions of
speciesjat a given value ofrdepends on the ion chargezjeand the electric potentialat
that position. The time-average electric potential in turn depends on the distribution of all
ions and is symmetric about the central ion, so expressions must be found forcj^0 andas
functions ofrthat are mutually consistent.