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and a theoretical energy efficiency
eq
¼
DG
o
DH
o
¼
1325
10
3
1
rev
1366
10
3
¼
97%
(11
:
6)
with F ¼ 96,485 C mol
21
the Faraday constant and n ¼ 12 the number of electrons
exchanged per molecule for complete oxidation to CO
2
.
However, under working conditions, with a current density j, the cell voltage E( j )
decreases greatly as the result of three limiting factors: the charge transfer overpoten-
tials h
a,act
and h
c,act
at the two electrodes due to slow kinetics of the electrochemical
processes (h
i
is defined as the difference between the working electrode potential E( j )
i
and the equilibrium potential E
eq,i
), the ohmic drop R
e
j
j
j
, with R
e
the ohmic resistance
of the electrolyte and interface, and the mass transfer limitations for reactants and
products. The cell voltage can thus be expressed as
E(
j
j
j
)
¼
E
2
(
j
j
j
)
E
1
(
j
j
j
)
¼
E
2
þ
h
c
(E
1
þ
h
a
)
R
e
j
j
j
¼
E
eq
(
j
h
a
jþj
h
c
jþ
R
e
j
j
j
)
(11
:
7)
where the overvoltages h
a
(.0 for an anodic reaction, i.e., here the oxidation of the
fuel) and h
c
(,0 for a cathodic reaction, i.e., here the reduction of the oxidant) take
into account both the slow rate of the electrochemical reactions (activation polariz-
ation) and the limiting rate of mass transfer (concentration polarization). Thus, the
energy efficiency can be expressed as
nFE
eq
DH
o
¼
n
exp
FE( j)
DH
o
¼
n
exp
n
E( j)
E
eq
1
C
2
H
5
OH
=
O
2
cell
¼
n
exp
n
DG
o
DH
o
¼
1
F
1
E
1
rev
E( j)
E
eq
(11
:
8)
eq
Therefore, for a DEFC working at 0.5 V at 50 mA cm
22
and leading either to CO
2
(complete oxidation) or to acetic acid (partial oxidation), the overall efficiency
would be
¼
12
12
0
:
5
1
C
2
H
5
OH
=
O
2
cell
1
:
144
0
:
97
42
:
4%
(11
:
9)
¼
4
12
0
:
5
1
C
2
H
5
OH
=
O
2
cell
1
:
144
0
:
97
14%
(11
:
10)
respectively. Since 1
eq
rev
is given by the thermodynamics (one can increase it slightly by
changing the operating pressure and temperature conditions), this expression shows
that the only ways to increase significantly the overall energy efficiency are to increase
1
E
, i.e., to decrease the overvoltages h, the ohmic drop R
e
j
j
j
, and to increase 1
F
, i.e., to
favor the complete oxidation of alcohol to CO
2
(avoiding the formation of aldehyde
and carboxylic acid). The decrease in
j
h
j
is directly related to the increase in the
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