Physical Chemistry , 1st ed.

EE° 

R

nF

T

ln Q

E° 

R

nF

T

ln 





i

ja

a

j

i

(

(

r

p

e

r

a

o

c

d

t

s

s

)

)





i

j (8.55)

where we have redefined Q, the reaction quotient, as

Q





i

ja

a

j

i

(

(

r

p

e

r

a

o

c

d

t

s

s

)

)





i

j (8.56)

where ai(prods) and aj(reacts) are the activitiesof the product and reactant
species, respectively. The exponents iand jare the stoichiometric coefficients
of the products and reactants, respectively, from the balanced chemical equa-
tion. The values for in Table 8.3 suggest that as ionic solutions become more
concentrated, properties like Efor an electrochemical reaction are less accu-
rately predicted using concentrations but more accurately predicted using ac-
tivities. The following example illustrates the difference.

Example 8.12

a.Approximate the expected voltage for the following electrochemical reac-

tion using the given molal concentrations.

2Fe (s) 3Cu^2 (aq, 0.050 molal) →2Fe^3 (aq, 0.100 molal) 3Cu (s)

b.Again approximate the expected voltage, but this time use the calculated

activities according to the Debye-Hückel theory.

The reaction occurs at 25.0°C. The value for Bat this temperature is

2.32    109 m^1 molal^ 1/2.Ais still 1.171 molal^ 1/2.

Assume that the molal concentrations are close enough to molar concen-

trations that they can be used directly. Additionally, assume that the anion is

NO 3

 , that is, that we are in reality considering 0.050-molal Cu(NO 3 ) 2 and

0.100-molal Fe(NO 3 ) 3 solutions. Also, use the fact that the average ionic radii

for Fe^3 and Cu^2 are 9.0 Å and 6.0 Å, respectively.

Solution

Using Table 8.2, we can easily determine that E°0.379 V and that the num-

ber of electrons transferred in the course of the molar reaction is 6.

a.Using the molal concentrations in the Nernst equation:

E0.379 V ln 

(

(

0

0

.

.

0

1

5

)

)

2

 3

E(0.379 0.00188) V 0.377 V

b.If, however, we use the Debye-Hückel formula, we first have to calculate

the activity coefficients of the ions:

ln Fe^3 

where we have converted the ionic radius of Fe^3 to units of meters and have

used the calculated ionic strength of a 0.100-molal Fe(NO 3 ) 3 solution. We get

ln Fe^3 

3.119

Fe^3 0.0442

1.171 molal^ 1/2( 3)^2 (0.600 molal)1/2



1 2.32  109 m^1 molal^ 1/29.00    10

10 m (0.600 molal)1/2

(8.314 moJlK)(298 K)



(6 mol e^ )(96,485 moCle )

8.7 Debye-Hückel Theory of Ionic Solutions 233