Solutions to exercises
77
4. Resistivity, conductivity, dielectric constant, permittivity, and relaxation
frequency are all characteristics that are theoretically independent of sample
geometry.
5. Given the results of the analysis of residues and the agreement between the
calculated points and the observations (see fig. 22), the most appropriate
circuit is the R //CPE circuit.
Solution 2.3 – Measurement of electronic conductivity
in a mixed conductor
1. The scheme for the electrochemical chain is
O 2 , Pt / CuO-Cu 2 O / (MIEC) / Pt, O 2
2. At the electrodes, the reactions are
O ( )
g
WE
2
1 2
+ 2e Pt m O
2−
MIEC
3. The equilibrium conditions of the system at the reference electrode are
2
1
μ
RE
O 2 + 2μ ˜
RE
e = μ ˜
RE
O 2−
and at the working electrode,
2
~
~
O
WE
e
WE
O
WE
2
1
2
2
μ
μ
μ
+
=
−
The equilibrium conditions between the various phases allow us to write
~
~
O
WE
O
RE
2
2
μ
μ
=
−
−
because J(O
2−
) = 0 (blocking electrode),
with
4F
WE
RE
O
WE
O
RE
2
2
ϕ
ϕ
μ
μ
−
=
−
^
`
h
j
and
U
4F
1
4F
1
RE
WE
O
WE
O
RE
O
2
2
2
ϕ
ϕ
μ
μ
μ
Δ
=
−
=
−
=
`
j
where φ
i
is the electrical potential of phase i.
Recalling that
μ
i
O 2 = μ
0,i
O 2 + RT ln a
i
O 2
we finally obtain
ln
U
F
RT
4
a
a
O
RE
O
WE
2
2
=
from which we deduce
a
a e
O
WE
O
RE RT
UF
2
2
4
=
4. a. At 750 °C, the oxygen activity at the working electrode is a
a e
O
WE
O
RE RT
UF
2
2
4
=