The Warburg coefficient (or Warburg constant),
A
W
, is the diffusion coefficient of ions in solution, associated to the Warburg element,
Z
W
. The Warburg coefficient,
A
W
, also written as,
σ
, has the units of
Ω
/
s
e
c
o
n
d
s
=
Ω
(
s
−
1
/
2
)
The value of
A
W
can be obtained by the gradient of the Warburg plot, a linear plot of the real impedance (
R
) against the reciprocal of the square root of the frequency (
1
/
ω
). This relation should always yield a straight line, as it is unique for a Warburg.
Alternatively, the value of
A
W
can be found by:
A
W
=
R
T
A
n
2
F
2
2
(
1
D
O
1
/
2
C
O
b
+
1
D
R
1
/
2
C
R
b
)
=
R
T
A
n
2
F
2
Θ
C
2
D
where
R
is the ideal gas constant,
T
is the thermodynamic temperature,
F
is the Faraday constant,
n
is the valency,
D
is the diffusion coefficient of the species where subscripts
O
and
R
stand for the oxidized and reduced species respectively,
C
b
is the concentration of the
O
and
R
species in the bulk, C is the concentration of the electrolyte,
A
denotes the surface area and
Θ
denotes the fraction of the
R
and
O
species present.
The equation for
A
W
applies to both reversible and quasi-reversible reactions for which both halves of the couple are soluble.
