Thermodynamics and Chemistry

CHAPTER 10 ELECTROLYTE SOLUTIONS

10.4 THEDEBYE–HUCKEL ̈ THEORY 296

0 0:02 0:04 0:06 0:08

0:7

0:8

0:9

0:10

1:0

mB=mol kg^1

̇

Figure 10.3 Mean ionic activity coefficient of aqueous HCl at 25 C. Solid curve:

experiment;a dashed curve: Debye–Huckel theory with ̈ a D 5  10 ^10 m; dotted

curve: Debye–Huckel limiting law. ̈

aRef. [ 74 ], Table 11-5-1.

When the solvent is water at 25 C, the quantitiesADHandBDHhave the values

ADHD1:1744kg1=2mol1=2 (10.4.5)

BDHD3:285 109 m^1 kg1=2mol1=2 (10.4.6)

From Eqs.10.3.8and10.4.1and the electroneutrality conditionCzCDz, we ob-
tain the following expression for the logarithm of the mean ionic activity coefficient of an
electrolyte solute:

ln (^) D

ADH

(^) zCz
p
Im
1 CBDHa
p
Im

(10.4.7)

In this equation,zCandzare the charge numbers of the cation and anion of the solute.
Since the right side of Eq.10.4.7is negative at finite solute molalities, and zero at infinite

dilution, the theory predicts that (^) is less than 1 at finite solute molalities and approaches
1 at infinite dilution.
Figure10.3shows that with the proper choice of the parametera, the mean ionic activity
coefficient of HCl calculated from Eq.10.4.7(dashed curve) agrees closely with experiment
(solid curve) at low molalities.
As the molalities of all solutes become small, Eq.10.4.7becomes
ln (^) DADH
(^)
zCz
p
Im (10.4.8)
(infinite dilution)
This form is known as theDebye–Huckel limiting law ̈. Note that the limiting law contains
no adjustable parameters. The dotted curve in Fig.10.3shows that the limiting law agrees
with experiment only at quite low molality.