The entropy maximization principle requires that at equilibrium
0 ¼ dS ¼
@S
1
ð Þ
@U 1
ð Þ
dU
1
ð Þ
þ
@S
1
ð Þ
@V 1
ð Þ
dV
1
ð Þ
þ
@S
2
ð Þ
@U 2
ð Þ
dU
2
ð Þ
þ
@S
2
ð Þ
@V 2
ð Þ
dV
2
ð Þ
By the closure conditions,
dU
2
ð Þ
¼ ÀdU
1
ð Þ
dV
2
ð Þ
¼ ÀdV
1
ð Þ
whence
0 ¼
1
T 1
ð Þ
À
1
T 2
ð Þ
equili
dU
1
ð Þ
þ
p
1
ð Þ
T 1
ð Þ
À
p
2
ð Þ
T 2
ð Þ
equili
dV
1
ð Þ
As this equation must hold independently for arbitrary values of dU
(1) and dV
(1) , we
must have
0.0000
0.9583
6200.0
6400.0
6600.0
6800.0
7000.0
7200.0
7400.0
7600.0
7800.0
8000.0
8200.0
U
(1) [J]
8017.071
Equilibrium
Initial Condition
S S (1) (250)
– S
(2) (350)
–
Fig. 9.2 Quasi-static trajectory toward internal mechanical equilibrium: total entropy versus
internal energy of the first subsystem
9.6 Thermal Equilibrium and Mechanical Equilibrium
257