Fundamentals of Electrochemical Double-Layer Supercapacitors
55
in Figure 2.3. This electrode potential is composed of the potential drops of
both the Helmholtz and diffuse layers. However, this potential drop is not
experimentally measurable. This can be illustrated and explained by the electrochemical measurement system shown in Figure 2.11. The measurement
for the potential drop across an interface such as the interface between metal
phase M I and the solution phase S must involve at least four interfaces such
as between the metal phase M I and the metal phase M II ( Δφ
II
M M
I /
II ), M and
solution phase S (Δφ II / ), S and M III
M S
( Δφ S M
II ), and M III and M I’ ( Δφ M
III
I′ ).
/
/M
If
the reading in the voltage meter is V (note V is the cell voltage), then V can be
expressed as:
V = Δφ M M
I /
II + Δ I
′
φ M
I /
+ Δφ
S
S/ M
III + Δφ M
III /M
I
(2.22)
Equation (2.22) suggests that it is impossible to measure Δφ M
II /S
in the presence of three uncertain interface potential drops unless they are fixed.
Equation (2.22) can be simplified to:
V = φ
′
M
I − φ M
I
(2.23)
where both φ M
I and φ ′
M
I are the inner potentials of the two lead metals.
According to Equation (2.1), Equation (2.23) can be alternatively expressed as:
V = ( ψ M
I + χ
I ′
M
I ) − ( ψ M
I′ − χ M )
(2.24)
For the same metal lead M I , χ M
I = χ M
I′ , Equation (2.24) can be converted into:
V = ψ M
I − ψ I ′ = Δψ I II + Δψ M
I
M
M / M
I /S
+ Δψ S/ M
III + Δψ III ′
M / /M
I
(2.25)
Comparing Equations (2.22) and (2.25), it can be seen that the sum difference
in the inner potential differences of the two electrodes is equal to that of the
outer potential differences.
In Equation (2.25), both Δψ I II
III
I′
M M
/
and Δψ M / M are metal–metal interfaces,
so their potential drops should be fixed as long as the types of metal are
fixed. If the value of Δψ
II
S M
/
III is fixed, the relative value of Δψ M /S can be
measured:
Δψ M
II S
V
/
= − ( Δψ III + Δψ
S/ M
M
I / M
II + Δψ M
III /M
I ) = −
V V ref r
(2.26)
where V is the cell voltage, and V ref is the electrode potential of the reference
electrode. Therefore, although the absolute potential drop across the interface
is not measureable, its relative value can be measured. In electrochemistry,
55
in Figure 2.3. This electrode potential is composed of the potential drops of
both the Helmholtz and diffuse layers. However, this potential drop is not
experimentally measurable. This can be illustrated and explained by the electrochemical measurement system shown in Figure 2.11. The measurement
for the potential drop across an interface such as the interface between metal
phase M I and the solution phase S must involve at least four interfaces such
as between the metal phase M I and the metal phase M II ( Δφ
II
M M
I /
II ), M and
solution phase S (Δφ II / ), S and M III
M S
( Δφ S M
II ), and M III and M I’ ( Δφ M
III
I′ ).
/
/M
If
the reading in the voltage meter is V (note V is the cell voltage), then V can be
expressed as:
V = Δφ M M
I /
II + Δ I
′
φ M
I /
+ Δφ
S
S/ M
III + Δφ M
III /M
I
(2.22)
Equation (2.22) suggests that it is impossible to measure Δφ M
II /S
in the presence of three uncertain interface potential drops unless they are fixed.
Equation (2.22) can be simplified to:
V = φ
′
M
I − φ M
I
(2.23)
where both φ M
I and φ ′
M
I are the inner potentials of the two lead metals.
According to Equation (2.1), Equation (2.23) can be alternatively expressed as:
V = ( ψ M
I + χ
I ′
M
I ) − ( ψ M
I′ − χ M )
(2.24)
For the same metal lead M I , χ M
I = χ M
I′ , Equation (2.24) can be converted into:
V = ψ M
I − ψ I ′ = Δψ I II + Δψ M
I
M
M / M
I /S
+ Δψ S/ M
III + Δψ III ′
M / /M
I
(2.25)
Comparing Equations (2.22) and (2.25), it can be seen that the sum difference
in the inner potential differences of the two electrodes is equal to that of the
outer potential differences.
In Equation (2.25), both Δψ I II
III
I′
M M
/
and Δψ M / M are metal–metal interfaces,
so their potential drops should be fixed as long as the types of metal are
fixed. If the value of Δψ
II
S M
/
III is fixed, the relative value of Δψ M /S can be
measured:
Δψ M
II S
V
/
= − ( Δψ III + Δψ
S/ M
M
I / M
II + Δψ M
III /M
I ) = −
V V ref r
(2.26)
where V is the cell voltage, and V ref is the electrode potential of the reference
electrode. Therefore, although the absolute potential drop across the interface
is not measureable, its relative value can be measured. In electrochemistry,