Physical Chemistry , 1st ed.

logarithm of. However, you may have to apply appropriate conversions in

order for the units to cancel properly.

How well do these equations work? First, we will consider equation 8.50, the

simplified Debye-Hückel limiting expression. The experimental values for

for 0.001-molal HCl and CaCl 2 at 25°C are 0.966 and 0.888, respectively.

The ionic strengths of the two solutions are 0.001 mand 0.003 m. In aqueous

solution, the value for Ais

A(2NAsolv)1/2

4 

e

0

2

rkT



3/2

 2 6.02     1023 mol^1  997 

m

kg

 3 )

1/2



3/2

where we have used the density of water as 997 kg/m^3 at 25°C and a dielectric

constant of 78.54, and the rest of the variables are fundamental constants that

can be obtained from tables.

Ultimately, the units work out to kg1/2/mol1/2, which is the reciprocal of the

square root of the molality unit, (molal)^ 1/2. Numerically, the overall value of

Acomes out as

A1.171 molal^ 1/2 (8.54)

(This value ofAis good for any aqueous solution at 25°C.) For HCl, in which

z 1 and z 

1, we have

ln (1.171 molal^ 1/2)  1 

 1  0.001 m olal

Notice how the square root of the molal units cancel. Numerically we have

ln 

0.03703

Therefore,

0.964

This value is very close to the experimental value of 0.966. For CaCl 2 , we have

ln (1.171 molal^ 1/2)  2 

 1  0.003 m olal 

0.1283

0.880

which is again very close to the experimental value of 0.888. Even the simple

form of the Debye-Hückel limiting law works very well for dilute solutions.

The more precise expression for the Debye-Hückel law is really necessary only

for more concentrated solutions.

Using Debye-Hückel theory, we can determine the activity coefficients of

ionic solutions. From these activity coefficients, we can determine the activi-

ties of ions in a solution. The activities of ions, in turn, are related to the mo-

lalities—that is, the concentrations—of ions in a solution. We must therefore

modify our approach in our understanding of the behavior of ionic solutions.

(Indeed, this idea applies to all solutions, but we are considering only ionic

solutions here.) Rather than relating the concentration of a solution to its

measurable properties, it is more accurate to relate the measurable properties

of an ionic solution to the activities of the ions.Thus, equations like equation

8.25 are better expressed as

(1.602 10

19 C)^2



4 8.854   10

12 

J

C

m

2

78.54 1.381  10

23 

K

J298 K

232 CHAPTER 8 Electrochemistry and Ionic Solutions