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introduced and others derivable from them, which constitute much of the useful
content of thermodynamic theory and are summarized in textbooks (for example
particular importance is the Gibbs-Duhem relation expressing an interdependence
among the intensive variables,
SdT
Vdp
þ
X
n
i
dl
i
¼
0
ð
2
:
7
Þ
In the application of the thermodynamic principles to a real physical situation,
it is necessary to know the explicit form of the fundamental relation, in any of its
equivalent versions (
2.1
)-(
2.6
), in order to fully express the physics of the sit-
uation, although the search for this explicit form is not strictly part of thermo-
dynamics itself. The physics of the situation can alternatively be introduced in
the form of equations of state, which are explicit relations between the inde-
pendent extensive variables and the intensive variables, such as the ideal gas
laws V
¼
nRT
=
p and U
¼
3nRT
=
2(R
¼
Lk).
Since systems at equilibrium are homogeneous within regions free from internal
walls it is sufficient to discuss the total amounts S
;
U
;
V
;
n
i
;
G
;
...... of the
extensive properties within the homogeneous regions. However, in chemical
thermodynamics, it is often useful to normalize the extensive properties to the total
amount of substance n (
¼
Rn
i
) in the system to give, respectively, the molar
quantities S
m
;
U
m
;
V
m
;
x
i
;
G
m
;
...... (where S
m
¼
S
=
n...and x
i
= mole fraction
of the i'th component); further, the molar quantities may be partitioned among the
components of substance as the partial molar quantities, noting that the partial
molar Gibbs energy is identical to the chemical potential.
2.3 Non-Equilibrium Thermodynamics
We now turn to systems out of equilibrium. Here, the thermodynamic treatment
rests on less well-defined foundations than does classical equilibrium thermody-
namics. There exist, in fact, a number of distinct approaches of diverse aims, but
two major branches of theory can be distinguished, characterized by Germain
(
1974
) as the ''ambitious attitude'' and the ''cautious attitude''. The former, var-
iously labeled as rational thermodynamics or continuum thermodynamics, claims
to attempt the broadest possible analysis. It aims not to depend on generalization
from classical equilibrium thermodynamics but to deal ab initio with processes
(described by constitutive relations) rather than with states. The physical specifi-
cation of the system at any instant involves not only the values of the measurable
parameters at that instant but also their histories at all previous instants, expressed
as functionals such as hereditary integrals (Malvern
1969
, pp. 256, 319). The
concepts of temperature and entropy are introduced as primitive quantities. No
attempt will be made to expound this approach here (see Coleman
1964
; Day
1972
; Noll
1974
; Truesdell
1984
).
