The second example is the case of “temperature equalization spontaneity” within
two parts of an isolated thermal composite system: A composite system consists of
gaseous subsystem
(1) and subsystem
(2) (same gas in both) of equal mole numbers
(N
(1) = N
(2) = N), initially at p and (T initial )
(1) , and p and (T initial )
(2) , respectively.
Assume (Nc p )
(1) =(Nc p )
(2) =10 kJ=K (the total heat capacity of the composite
system is 20 kJ=K), (T initial )
(1) = 450 K and (T initial )
(2) = 200 K, and a special heat
reservoir at T 0 = 300 K. The spontaneous event is now a cooling event of the
composite system from an average temperature of 325 to 300 K (in both subsystems). The corresponding heat exchange is
Q spon ¼ À Nc p
À
Á ð1Þ 450 À 300
ð
Þþ Nc p
À
Á ð2Þ 200 À 300
ð
Þ
h
i
¼ À500 kJ
We note that this is an arbitrarily constructed spontaneous event, not the normal
spontaneous event of an isolated system that interacts with no heat reservoir (as
considered again in example below).
The reversible event is an isentropic event of the composite system with a
“silent” or “non-participating” heat reservoir. That is,
Q rev ¼ 0 kJ
Thus,
D ^
Q ¼ 0 À À500
ð
Þ¼500 kJ
The isentropic event is brought about by a Carnot heat engine going through a
sequence of micro-cycles: During each micro-cycle the Carnot engine extracts
T
1
ð Þ
T 2
ð Þ dQ
2
ð Þ from subsystem 1 and rejects dQ
2
ð Þ amount of heat to subsystem 2. dQ
2
ð Þ
is related to the temperature change of subsystem 2, dQ
2
ð Þ
¼ Nc p dT
2
ð Þ , and changes
in T
(1) and T
(2) are correlated according to T
(1)
 T
(2) = 450 Â 200. It can be
readily shown, noting that for an isentropic event T
1
ð Þ
Á T
2
ð Þ
¼ 300
2
W Rev ¼
Z
dW ¼
Z
dQ
1
ð Þ
À dQ
2
ð Þ
¼
Z 300
T 2
ð Þ ¼200
10dT
2
ð Þ T
1
ð Þ
T 2
ð Þ
À 1
¼ D ^
Q ¼ 500kJ
The schematic heat relations are shown in Fig. 8.5.
This second example also represents a case of converting heat 100% into work.
In this case, heat appears to be from the source system of the thermal composite
body. But, the above treatment, though correct, is not satisfactory since it neither
conforms to neither Eq. (123) nor Eq. (130). It is misleading to view this to be a
case of transformation of change in the form of energy of the source system into
work. Instead, it should be viewed like the free expansion example to be a case of
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8 The Second Law: The Entropy Growth Potential Principle …
two parts of an isolated thermal composite system: A composite system consists of
gaseous subsystem
(1) and subsystem
(2) (same gas in both) of equal mole numbers
(N
(1) = N
(2) = N), initially at p and (T initial )
(1) , and p and (T initial )
(2) , respectively.
Assume (Nc p )
(1) =(Nc p )
(2) =10 kJ=K (the total heat capacity of the composite
system is 20 kJ=K), (T initial )
(1) = 450 K and (T initial )
(2) = 200 K, and a special heat
reservoir at T 0 = 300 K. The spontaneous event is now a cooling event of the
composite system from an average temperature of 325 to 300 K (in both subsystems). The corresponding heat exchange is
Q spon ¼ À Nc p
À
Á ð1Þ 450 À 300
ð
Þþ Nc p
À
Á ð2Þ 200 À 300
ð
Þ
h
i
¼ À500 kJ
We note that this is an arbitrarily constructed spontaneous event, not the normal
spontaneous event of an isolated system that interacts with no heat reservoir (as
considered again in example below).
The reversible event is an isentropic event of the composite system with a
“silent” or “non-participating” heat reservoir. That is,
Q rev ¼ 0 kJ
Thus,
D ^
Q ¼ 0 À À500
ð
Þ¼500 kJ
The isentropic event is brought about by a Carnot heat engine going through a
sequence of micro-cycles: During each micro-cycle the Carnot engine extracts
T
1
ð Þ
T 2
ð Þ dQ
2
ð Þ from subsystem 1 and rejects dQ
2
ð Þ amount of heat to subsystem 2. dQ
2
ð Þ
is related to the temperature change of subsystem 2, dQ
2
ð Þ
¼ Nc p dT
2
ð Þ , and changes
in T
(1) and T
(2) are correlated according to T
(1)
 T
(2) = 450 Â 200. It can be
readily shown, noting that for an isentropic event T
1
ð Þ
Á T
2
ð Þ
¼ 300
2
W Rev ¼
Z
dW ¼
Z
dQ
1
ð Þ
À dQ
2
ð Þ
¼
Z 300
T 2
ð Þ ¼200
10dT
2
ð Þ T
1
ð Þ
T 2
ð Þ
À 1
¼ D ^
Q ¼ 500kJ
The schematic heat relations are shown in Fig. 8.5.
This second example also represents a case of converting heat 100% into work.
In this case, heat appears to be from the source system of the thermal composite
body. But, the above treatment, though correct, is not satisfactory since it neither
conforms to neither Eq. (123) nor Eq. (130). It is misleading to view this to be a
case of transformation of change in the form of energy of the source system into
work. Instead, it should be viewed like the free expansion example to be a case of
206
8 The Second Law: The Entropy Growth Potential Principle …