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5 Controversy of Thermodynamics Associated with Surface …
to a solid electrode and for the surface stress measurement by a cantilever bending
method. The issues result from the lack of thermodynamic constraint such as constant
elastic strain or electrode potential. We review the above controversial arguments.
5.2 On Homogeneous Nature of the Thermodynamic
Functions of Solid Electrode
Láng and Heusler [1–3] have claimed that the excess internal energy of solid electrode U
σ is not a homogeneous function of the first order with respect to all variables. The excess internal energy of the electrode surface can be expressed as functions of surface entropy S
σ , surface electric charge Q
σ , mole numbers of all surface
components n
σ
1 , …, n
σ
m , and surface area A:
U
σ
= U
σ
(S
σ
, Q
σ
, n
σ
1 , . . . , n
σ
m , A).
(5.1)
If U
σ is a homogeneous function of the first order, the following relationship holds
for any real number λ > 0:
U
σ
(λS
σ
, λQ
σ
, λn
σ
1 , . . . , λn
σ
m , λA)
= λU
σ
(S
σ
, Q
σ
, n
σ
1 , n
σ
m , A).
(5.2)
Its differentiation with respect to λ provides
∂U
σ
λS
σ
, λQ
σ
, λn
σ
1 , . . . ., λn
σ
m , λA
∂(λS σ )
∂(λS
σ
)
∂λ
+
∂U
σ
λS
σ
, λQ
σ
, λn
σ
1 , . . . ., λn
σ
m , λA
∂(λQ σ )
∂(λQ
σ
)
∂λ
+
∂U
σ
λS
σ
, λQ
σ
, λn
σ
1 , . . . ., λn
σ
m , λA
∂
λn
σ
1
∂
λn
σ
1
∂λ
+ . . . +
∂U
σ
λS
σ
, λQ
σ
, λn
σ
1 , . . . ., λn
σ
m , λA
∂
λn σ
m
∂
λn
σ
m
∂λ
+
∂U
σ
λS
σ
, λQ
σ
, λn
σ
1 , . . . ., λn
σ
m , λA
∂(λA)
∂(λA)
∂λ
= U
σ
S
σ
, Q
σ
, n
σ
1 , . . . , n
σ
m , A
.
(5.3)
In the case of λ = 1, Eq. (5.3) takes the form:
∂U
σ
∂S σ S
σ
+
∂U
σ
∂Q σ Q
σ
+
∂U
σ
∂n
σ
1
n
σ
1 + . . . +
∂U
σ
∂n σ
m
n
σ
m +
∂U
σ
∂A
A = U
σ
.
(5.4)