combined system even if the individual systems are not isolated. This principle is a
powerful instrument of drawing quite general inference from the second law. Note,
however, valid and useful inference can only be drawn in cases that such a system,
when it is “defined” to be isolated, remains meaningful, not when the system has its
fundamental characteristic changed when it is so defined. For example, the planet
Earth would lose its meaning as a system if it is so defined!
The fact that all spontaneous transformations in an isolated system proceed in
such a direction dictated by (74) can be illustrated by two simple examples. As the
first example, we consider a dispersal irreversible process of a composite system,
the exchange of heat by heat conduction between two bodies, A 1 and A 2 . Let T 1 and
T 2 be the temperatures of these two bodies, respectively, and let T 1 \ T 2 . Since
heat flows by conduction from the hotter body to the colder body, the hotter body
A 2 gives up a quantity of heat dQ which is absorbed by A 1 . Thus, the entropy of A 1
changes by an amount of dQ=T 1 , while that of A 2 by an amount of ÀdQ=T 2 . The
entropy change of the isolated composite system is, accordingly,
dQ
T 1
À
dQ
T 2
Since T 1 \ T 2 , this change of entropy for the isolated composite system,
dQ
T 1
À
dQ
T 2
[ 0
ð
Þ, is positive.
Now, let us consider the continuing heat exchange between the two parts
eventually leading to thermal equilibrium between A 1 and A 2, Suppose changes in
the temperatures of A 1 and A 2 are related to heat exchange by
dQ ¼ C p1 dT 1
dQ ¼ ÀC p2 dT 2
The final temperature at thermal equilibrium, therefore, is
T final ¼
C p1 T 1 þ C p2 T 2
C p1 þ C p2
¼
T 1 þ C p2
C p1
À
Á
T 2
1 þ C p2
C p1
Correspondingly, the total entropy change of the composite system is (assuming
pressure remaining constant and constant C
0
p s) obtained from the application of
Eq. (63C) to each part
DS ¼ C p1 ln
T final
T 1
þ C p2 ln
T final
T 2
¼ C p1 ln
T final
À
Á 1 þ C p2 =Cp1
T 1 T 2
ð Þ
C p2= C 1
h
i
This quantity can be shown to be positive, consistent with the entropy principle.
5.5 The Principle of the Increase of Entropy
103
powerful instrument of drawing quite general inference from the second law. Note,
however, valid and useful inference can only be drawn in cases that such a system,
when it is “defined” to be isolated, remains meaningful, not when the system has its
fundamental characteristic changed when it is so defined. For example, the planet
Earth would lose its meaning as a system if it is so defined!
The fact that all spontaneous transformations in an isolated system proceed in
such a direction dictated by (74) can be illustrated by two simple examples. As the
first example, we consider a dispersal irreversible process of a composite system,
the exchange of heat by heat conduction between two bodies, A 1 and A 2 . Let T 1 and
T 2 be the temperatures of these two bodies, respectively, and let T 1 \ T 2 . Since
heat flows by conduction from the hotter body to the colder body, the hotter body
A 2 gives up a quantity of heat dQ which is absorbed by A 1 . Thus, the entropy of A 1
changes by an amount of dQ=T 1 , while that of A 2 by an amount of ÀdQ=T 2 . The
entropy change of the isolated composite system is, accordingly,
dQ
T 1
À
dQ
T 2
Since T 1 \ T 2 , this change of entropy for the isolated composite system,
dQ
T 1
À
dQ
T 2
[ 0
ð
Þ, is positive.
Now, let us consider the continuing heat exchange between the two parts
eventually leading to thermal equilibrium between A 1 and A 2, Suppose changes in
the temperatures of A 1 and A 2 are related to heat exchange by
dQ ¼ C p1 dT 1
dQ ¼ ÀC p2 dT 2
The final temperature at thermal equilibrium, therefore, is
T final ¼
C p1 T 1 þ C p2 T 2
C p1 þ C p2
¼
T 1 þ C p2
C p1
À
Á
T 2
1 þ C p2
C p1
Correspondingly, the total entropy change of the composite system is (assuming
pressure remaining constant and constant C
0
p s) obtained from the application of
Eq. (63C) to each part
DS ¼ C p1 ln
T final
T 1
þ C p2 ln
T final
T 2
¼ C p1 ln
T final
À
Á 1 þ C p2 =Cp1
T 1 T 2
ð Þ
C p2= C 1
h
i
This quantity can be shown to be positive, consistent with the entropy principle.
5.5 The Principle of the Increase of Entropy
103