Thermodynamics and Chemistry

CHAPTER 10 ELECTROLYTE SOLUTIONS

10.5 DERIVATION OF THEDEBYE–HUCKEL ̈ EQUATION 298

HCl

CaCl 2

0 0:1 0:2

0:5

0:4

0:3

0:2

0:1

0:3

0

Im=mol kg^1


1=2

ln

̇

Figure 10.4 Dependence of ln (^) on
p
Imfor aqueous HCl (upper curves) and aque-
ous CaCl 2 (lower curves) at 25 C.a Solid curves: experimental; dashed curves:
Debye–Huckel equation ( ̈ aD 5  10 ^10 m for HCl,aD4:5 10 ^10 m for CaCl 2 );
dotted lines: Debye–Huckel limiting law. ̈
aExperimental curves from parameter values in Ref. [ 74 ], Tables 11-5-1 and 12-1-3a.
Debye and Huckel assumed that ̈ cj^0 is given by the Boltzmann distribution
cj^0 Dcjezje=kT (10.5.1)
wherezjeis the electrostatic energy of an ion of speciesj, andkis the Boltzmann con-
stant (kDR=NA). Asrbecomes large,approaches zero andcj^0 approaches the macro-
scopic concentrationcj. AsT increases,cj^0 at a fixed value ofrapproachescj because
of the randomizing effect of thermal energy. Debye and Huckel expanded the exponential ̈
function in powers of1=T and retained only the first two terms:cj^0 cj.1zje=kT /.
The distribution of each ion species is assumed to follow this relation. The electric potential
function consistent with this distribution and with the electroneutrality of the solution as a
whole is
D.zie=4r 0 r/e.ar/=.1Ca/ (10.5.2)
Hereis defined by^2 D2NA^2 e^2 Ic=r 0 RT, whereIcis theionic strength on a concen-
tration basisdefined byIcD.1=2/

P

iciz
2
i.
The electric potentialat a point is assumed to be a sum of two contributions: the
electric potential the central ion would cause at infinite dilution,zie=4r 0 r, and the
electric potential due to all other ions,^0. Thus,^0 is equal tozie=4r 0 r, or

^0 D.zie=4r 0 r/åe.ar/=.1Ca/1ç (10.5.3)

This expression for^0 is valid for distances from the center of the central ion down toa, the
distance of closest approach of other ions. At smaller values ofr,^0 is constant and equal
to the value atrDa, which is^0 .a/D.zie=4r 0 /=.1Ca/. The interaction energy