initially. After the valve opens, the composite system eventually reaches
thermal equilibrium internally. Assuming constant c V and c p , the final
equilibrium temperature, T m , will be
T m ¼
T 1 þ T 2
2
Show that the total entropy change of the composite system is
DS ¼ Nc p ln
T
2
m
T 1 T 2
¼ Nc p ln
ðT 1 þ T 2 Þ
2
4T 1 T 2
:
Explain why the composite system entropy change equals the process entropy production DS ¼ ðD P SÞ spon .
8:10 Consider the second example in Sect. 8.4 as shown in Fig. 8.5: Assume Nc p
= 10 kJ/K (the total heat capacity of the composite system is 20 kJ/K),
T 1 = 450 K and T 2 = 200 K, the temperature of the reservoir T 0 = 325 K.
Calculate (D P S) spon . Prescribe in terms of theoretical operating steps how
one can convert the spontaneous change into a reversible change between
the same end states and show that the reversible work obtained is equal to
T 0 D P S
ð
Þ spon :
Hint: The theoretical operating steps will be a two-stage operation––the first
stage, 1 ! 2S, is an isentropic stage of a Carnot engine operating between
the two subsystems. Because it is an isentropic stage
T
ð1Þ
¼
450 Â 200
T ð2Þ
and T
ð1Þ
¼ T
ð2Þ
¼ T 2S ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
450 Â 200
p
¼ 300K
Correspondingly, the Carnot engine produces
W 1!2S ¼
Z
dW ¼
Z
300 K
200 K
Nc p
T
ð1Þ
T ð2Þ À 1
!
dT
ð2Þ
The second stage, 2S ! 2, is a heating of the composite system/thermal
mass toward 325 K.
230
8 The Second Law: The Entropy Growth Potential Principle …