same usefulness as a fundamental function. In these cases, for the purpose of
preserving the all-inclusive information, the dependent variable U can be transformed by Legendre transformation [1:137–149] into a new variable, i.e., an
alternative fundamental functional variable, w.
Consider the case of the transformation of U-S-V into w-S-p, in which w is the
new dependent variable
w U À
@U
@V
S;N i
V ¼ U À Àp
ð ÞV ¼ U þ pV
ð100Þ
which has been identified as the enthalpy, H (Sect. 3.6).
Consider the case of the transformation of U-S-V into w-T-V
w U À
@U
@S
V;N i
S ¼ U À TS
ð100Þ
which will be called the Helmholtz function, A H .
Consider next the case of the transformation of A H -T-V into w-T-p, one gets
w ¼ G
ð
Þ¼ A H À
@A H
@V
T;N i
ÁV ¼ A H þ pV ¼ U À TS þ pV
ð101Þ
which will be called the Gibbs function, G.
The complete set of new dependent variables is summarized in Table 7.1.
7.1.1 The Extremum Principle for Thermodynamic
Equilibriums of Composite Systems
We now consider composite systems approaching internal equilibrium with the
possibility of being subject to the constraint of reservoir that the systems interact
with.
Table 7.1 Thermodynamic potentials
Independent
variables,
X and Y
Dependent thermodynamic functional
variable w derived from Legendre
transformation
Name of the
alternative
dependent
variable w
Fundamental
function,
w = w(X, Y)
S and V
U
U -S-V
S and p
U+ pV
Enthalpy H
H -S-p
T and V
U− TS
Helmholtz
function A H
A H -T-V
T and p
H– TS = A H + pV = U + pV − TS
Gibbs function
G
G-T-p
7.1 Thermodynamic Potentials and Free Energies
159