Equation (64) in Chap. 5 is reproduced here
dU ¼ TdS À pdV
ð64Þ
Equation (64) suggests that the set of U-S-V is a unique one. Their functional
relationship may be written as
f U; S; V
ð
Þ¼0
ð96Þ
Equation (64) is known under the name of fundamental differential or principal
exact differential. Correspondingly, Eq. (96) may be referred to as a fundamental
thermodynamic function of state in the sense that its differentiation results in Eq. (64).
The fundamental thermodynamic function and the fundamental differential,
which “capture the complete thermodynamic information about a system” (see
Chap. 9), serve as the foundation of equilibrium thermodynamics, which will be
treated in Chap. 9. But, it does not capture all aspects of the first and the second
laws, which are the foundation of engineering thermodynamics that studies the
interaction of a system and its surroundings, as it will be evident in the following
treatment of free energy and exergy. Even so, the concept of free energies begins
with the concept of thermodynamic potentials, which (the potential expressions
themselves) can be conveniently introduced in terms of equilibrium thermodynamics whereas the interpretation of thermodynamic potential as free energies must,
of course, involve how the system interacting with its surroundings.
Beginning with the energy representation of the fundamental function, Eq. (96),
U ¼ U S; V
ð
Þ
ð96AÞ
we compute the first differential
dU ¼
@U
@S
V
dS þ
@U
@V
S
dV
ð97Þ
Comparing Eq. (97) with Eq. (64)
T
@U
@S
V
¼ T S; V
ð
Þ
ð98Þ
Àp
@U
@V
S
¼ Àp S; V
ð
Þ
ð99Þ
Equations (96A), (98), and (99) are called canonical form, and S and V the
generalized coordinates of the canonical form.
It is useful to choose a different set of independent variables as generalized
coordinates, and the question arises whether the resulting U-function of the new set
of independent variables, e.g., S-p or T-V instead of the original S-V, retains the
158
7 Free Energy, Exergy, and Energy …