68
2 – Methods and techniques
c. Deduce the activation energies for the total conductivity and the cationic
conductivity. What can we conclude?
d. Give two examples of binary ionic crystals that have characteristics
similar to those of CaF 2 .
Exercise 2.7 – Electrochemical semipermeability
Consider a MX ionic crystal with the following chemical potential difference
for the metal M applied between its extremities (1) and (2):
.
M
(1)
M
(2)
μ
μ
−
1. Give the expression for the potential difference
a. E 1 2
a
Δ − for the case of a purely cationic conductor,
b. E 1 2
b
Δ − for the case of a mixed conductor and as a function of the average
cationic transport number
.
t M +
2. In the absence of an external electric current, derive the expression for the
gradient of the electrical potential, d φ, at an arbitrary point of a mixed
conductor and as a function of the transport number t M +, of the gradients
of the chemical potential,
M
d μ + , of the cations M
+
, and of the electrons
(d μ e ).
3. Deduce the expression of the particulate cationic current I M + as a function
of the electronic conductivity σ e , t M +, and d μ M .
4. Given that I M + is conserved, give the expression for the flux of species M
+
as a function of E
a
1 2
Δ − , of the total conductivity σ t , of the average value
(1
)
t
t
M
M
−
+
+ , and of the length ℓ of the sample.
Exercise 2.8 – Determination of transport number
by electrochemical semipermeability
We want to determine the electronic transport number of a mixed-conductor
solid solution (ionic by O
2−
ions and p-type electronic). When a pellet of this
compound separates two compartments T and CE containing gases with different
oxygen partial pressures, the pellet is traversed by an oxygen flux called the
specific flux J S,O 2 , which is expressed as
J
V
D
P
P S
S,O
m
t
2
,
Δ
=
#
#
(1)