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i
0
H
+
/H
2
is the exchange current of the redox reaction for hydrogen and
i
0
HPW
is the exchange current of the redox reaction for phosphotungstic acid.
β
H
is
the transfer coecient of the oxidation of hydrogen.
R
is the gas constant
and its value is 8
.
314
J
−
1
·
K
·
mol
−
1
.
F
is the Faraday constant and its value
is 96485
C
mol
−
1
.
α
HPW
is the transfer coecient of the reduction reaction of
phosphotungstic acid.
T
is the absolute temperature and the unit is
K
.
E
μ
is the
mixed potential.
E
HPW
is the balance potential of hydrogen, which conforms to
the below Nernst equation.
·
2
F
ln
[
H
+
]
2
E
H
+
/H
2
=
RT
(5)
pH
2
We can get the expression of the mixed potential
E
μ
by solving the equation
(3), (4) and (5):
(2
β
H
+
α
HPW
)
H
log
i
0
HPW
α
HPW
2
β
H
+
α
HPW
+
2
.
3
RT
E
μ
=
i
0
H
+
/H
2
log
[
H
+
]
2
pH
2
β
H
2
β
H
+
α
HPW
·
2
.
3
RT
F
+
(6)
In equation (6), [
H
+
] is the proton concentration of hydrogen.
p
H
2
is the pressure
of hydrogen, whose unit is atm.
E
HPW
is the balance potential for the reduction
reaction of phosphotungstic acid and can be expressed as:
PW
12
(
VI
)
O
3
−
40
PW
11
(
VI
)
W
(
V
)
O
3
−
E
HPW
=
E
0
HPW
+
RT
F
ln
(7)
40
In equation (7),
E
0
HPW
is the standard electrode potential of phosphotungstic
acid and it can be denoted by equation (8).
E
0
HPW
≈
298
.
15) (8)
In equation (8),
a
and
b
are the temperature coecients of standard electrode
potential of phosphotungstic acid, which are constant. In equation (6),
i
0
HPW
is
the exchange current density of the redox reaction of phosphotungstic acid and
it can be expressed as:
a
+
b
(
T
−
i
0
HPW
=2
Fk
HPW
[
PW
12
(
VI
)
O
3
−
40
]
β
HPW
[
PW
11
(
VI
)
W
(
V
)
O
3
−
40
]
α
HPW
·
(9)
k
HPW
is the standard reaction rate constant of the redox reaction of phospho-
tungstic acid.
We can conclude from equation (6) that the relationship between the mixed
potential and the hydrogen concentration can be described by half logarithmic
function. However the coecient in front of the logarithmic is not that of the
Nernst equation (59/n mV when 25
◦
C).
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