1.5 Major Parameters of Surface Thermodynamics
9
U
σ
= T
σ S
σ
+ γ A +
i
μ
σ
i n
σ
i .
(1.26)
If α, σ, and β phases are in thermodynamic equilibrium, T
σ
= T
α
= T
β
= T and
μ
σ
i = μ
α
i = μ
β
i = μ i hold so that superscripts of T and μ i are not needed. In the
bulk α and β phases, the following relationships hold:
U
α
= TS
σ
− PV
α
+
i
μ i n
α
i ,
(1.27)
and
U
β
= TS
β
− PV
β
+
i
μ i n
β
i .
(1.28)
The Helmholtz free energy F in the respective phase is obtained from the internal
energy U by Legendre transformation:
F
α
= U
α
− TS
α
= −PV
α
+
i
μ i n
α
i ,
(1.29)
F
β
= U
β
− TS
β
= −PV
β
+
i
μ i n
β
i ,
(1.30)
and
F
σ
= U
σ
− TS
σ
= γ A +
i
μ i n
σ
i .
(1.31)
Furthermore, division of Eq. (1.31) by surface area A leads to
γ = f
σ
−
i
Γ i μ i ,
(1.32)
where f
σ
=
F
σ
A
and Γ i =
n
σ
i
A
are the Helmholtz free energy per unit surface area and
the adsorption of component i, respectively. In the one component system, γ is equal
to f
σ by choosing the location of dividing surface at Γ 1 = 0. On the other hand, in
the multicomponent systems, γ is not equal to f
σ since Γ i =1 is not zero, even if the
location of dividing surface is chosen at Γ 1 = 0 [5]. Nevertheless, γ is independent
of the location of the dividing surface, while f
σ is dependent of the location.
The Gibbs free energy G in the respective phase is obtained from the internal
energy U, and the enthalpy H (= U + PV ) by two Legendre transformations:
G
α
= H
α
− TS
α
=
i
μ i n
α
i ,
(1.33)