Thermodynamics - Materials Science for Structural Geology - page 22

Only changes in the energy of a system can be measured and these changes derive
from work done on the system or from heat or substance added to the system. The
internal energy is thus an extensive variable of fundamental importance in
describing the state of the system. Other extensive or additive variables of similar
importance are the amount of substance in the system and the dimensions of the
system, if mechanical work is involved, or analogous parameters associated with
other forms of work (electrical, etc.). For simplicity in exposition we shall here
only consider systems having as the other extensive variables the volume V and
the amounts of substance (moles) n i of each of i components (n i ¼ N i = L where N i
is the number of molecules or entities of substance i and L is the Avogadro
number). The state of such a system is then completely specified by U, V and n i if
it is at equilibrium.
However, we also wish to consider systems in which changes of state (transi-
tions or processes) are occurring and to establish criteria of equilibria. Further, it is
well known that real physical processes are irreversible or dissipative in some
fundamental sense and we need a criterion for determining the direction of change.
In order to deal with these aspects, another extensive variable and function of state
called the entropy S is introduced through the Second Law, according to which, in
an isolated system, entropy is unchanged (DS ¼ 0) in a reversible process and
increases (DS [ 0) in an irreversible process. This law can be restated to give a
criterion of equilibrium, namely, that in an isolated system the entropy is a
maximum at equilibrium.
Some insight into the nature of entropy can be obtained from the molecular
point of view of statistical mechanics, in which entropy is given by S ¼ k ln g
where k is the Boltzmann constant and g is the number of quantum states acces-
sible to the system and assumed to be equally probable (Kittel and Kroener 1980 ,
Chap. 2 ) , that is, the number of different microscopic possibilities or configurations
under which the given thermodynamic state can be realized. Macroscopically, it
can only be stated that the entropy is a function of the other extensive variables,

S ¼ SU ; ð V ; n i Þ ð 2 : 1 Þ

This relation serves as a fundamental relation from which all other properties of
the thermodynamic system in equilibrium can be derived, since the specification of
the extensive variables fully characterizes the state of the system. From consid-
eration of the differential of S in the case of the reversible addition of an amount of
heat DQ to the system at constant V and n i it follows that DS ¼ DQ = T ,orS ¼

R DQ = T where the integration path is a reversible path; in the irreversible case,
DS [ DQ = T but no other statement can in general be made, that is, we cannot in
general define an entropy exactly in a system out of equilibrium unless some
restrictive statements are made about the nature of the system.
The fundamental relation in the ''entropic form'' ( 2.1 ) can be rewritten in the
''energetic form''

U ¼ US ; ð V ; n i Þ ð 2 : 2 Þ