6
1 Surface Thermodynamics of Solid Electrode
In Eqs. (1.9) and (1.10), n i , n 1 , c
α
i , c
β
i , c
α
1 , c
β
1 , and V are dependent only of the
state of the system and are independent of the location of the dividing surface. In
contrast, n
σ
i , n
σ
1 , and V
β depend on the location of the dividing surface. The following
relationship between n
σ
i and n
σ
1 [4] is derived by eliminating V
β between Eqs. (1.9)
and (1.10):
n
σ
i − n
σ
1
c
α
i − c
β
i
c
α
1 − c
β
1
=
n i − c
α
i V
−
n 1 − c
α
1 V
c
α
i − c
β
i
c
α
1 − c
β
1
.
(1.11)
The right-hand side of Eq. (1.11) consists only of quantities independent of the
location of the dividing surface. Consequently, the value of the left-hand side is
invariant, irrespective of the location of the dividing surface. The quantity of n
σ
i
divided by the dividing surface area A is represented by
Γ i =
n
σ
i
A
,
(1.12)
where Γ i is named “surface excess of component i.” The quantity of the left-hand
side divided by A in Eq. (1.11) is named “relative surface excess of component i with
respect to component 1,” and is denoted by Γ i,1 :
Γ i,1 = Γ i − Γ 1
c
α
i − c
β
i
c
α
1 − c
β
1
.
(1.13)
Since Γ i,1 is independent of the location of the dividing surface, even though both
Γ i and Γ 1 depend on the location, Γ i,1 for any arbitrary location z is given by
Γ i,1 = (Γ i ) z − (Γ 1 ) z
c
α
i − c
β
i
c
α
1 − c
β
1
.
(1.14)
In the case where the location of the dividing surface is chosen at Γ 1 = 0, Γ i,1 reduces
to
Γ i,1 = (Γ i ) z 0 .
(1.15)
Therefore, Γ i,1 is equivalent to Γ i at the dividing surface of Γ 1 = 0. This explains
why the application of the Gibbs model to the interface leads to the definition of
quantities which has a direct experimental significance.
Although Eq. (1.9) contains the terms of concentration and volume, n
σ
i or Γ i can
be expressed in terms of mole fraction:
n
σ
i = n i − n
α x
α
i − n
β x
β
i ,
(1.16)
1 Surface Thermodynamics of Solid Electrode
In Eqs. (1.9) and (1.10), n i , n 1 , c
α
i , c
β
i , c
α
1 , c
β
1 , and V are dependent only of the
state of the system and are independent of the location of the dividing surface. In
contrast, n
σ
i , n
σ
1 , and V
β depend on the location of the dividing surface. The following
relationship between n
σ
i and n
σ
1 [4] is derived by eliminating V
β between Eqs. (1.9)
and (1.10):
n
σ
i − n
σ
1
c
α
i − c
β
i
c
α
1 − c
β
1
=
n i − c
α
i V
−
n 1 − c
α
1 V
c
α
i − c
β
i
c
α
1 − c
β
1
.
(1.11)
The right-hand side of Eq. (1.11) consists only of quantities independent of the
location of the dividing surface. Consequently, the value of the left-hand side is
invariant, irrespective of the location of the dividing surface. The quantity of n
σ
i
divided by the dividing surface area A is represented by
Γ i =
n
σ
i
A
,
(1.12)
where Γ i is named “surface excess of component i.” The quantity of the left-hand
side divided by A in Eq. (1.11) is named “relative surface excess of component i with
respect to component 1,” and is denoted by Γ i,1 :
Γ i,1 = Γ i − Γ 1
c
α
i − c
β
i
c
α
1 − c
β
1
.
(1.13)
Since Γ i,1 is independent of the location of the dividing surface, even though both
Γ i and Γ 1 depend on the location, Γ i,1 for any arbitrary location z is given by
Γ i,1 = (Γ i ) z − (Γ 1 ) z
c
α
i − c
β
i
c
α
1 − c
β
1
.
(1.14)
In the case where the location of the dividing surface is chosen at Γ 1 = 0, Γ i,1 reduces
to
Γ i,1 = (Γ i ) z 0 .
(1.15)
Therefore, Γ i,1 is equivalent to Γ i at the dividing surface of Γ 1 = 0. This explains
why the application of the Gibbs model to the interface leads to the definition of
quantities which has a direct experimental significance.
Although Eq. (1.9) contains the terms of concentration and volume, n
σ
i or Γ i can
be expressed in terms of mole fraction:
n
σ
i = n i − n
α x
α
i − n
β x
β
i ,
(1.16)