For a thermally isolated composite system subject to constant system volume,
V ¼ V
1
ð Þ
þ V
2
ð Þ
¼ constant, its system internal energy U is also constant in
accordance with the first law. Consider the entropy representation of the fundamental function
S ¼ S U; V
ð
Þ¼S
1
ð Þ U
1
ð Þ
; V
1
ð Þ
þ S
2
ð Þ U
2
ð Þ
; V
2
ð Þ
ð96BÞ
Application of the entropy principle to Eq. (96B) leads to the extremum principle of maximum system entropy at internal thermodynamic equilibrium
S ¼ S
1
ð Þ U
1
ð Þ
; V
1
ð Þ
þ S
2
ð Þ U
2
ð Þ
; V
2
ð Þ
!S max
That is, the equilibrium values of U
1
ð Þ
equi ; U
2
ð Þ
equi ; V
1
ð Þ
equi ; V
2
ð Þ
equi are determined by
dS ¼ 0; at U
1
ð Þ
¼ U
1
ð Þ
equi and V
1
ð Þ
¼ V
1
ð Þ
equi
ð102AÞ
under the constraints
U
1
ð Þ
þ U
2
ð Þ
¼ U
1
ð Þ
initial þ U
2
ð Þ
initial ; a constant
V
1
ð Þ
þ V
2
ð Þ
¼ V
1
ð Þ
initial þ V
2
ð Þ
initial ; a constant
Consider a thermally isolated composite system in interaction with a
constant-pressure reservoir, i.e., p
1
ð Þ
¼ p
2
ð Þ
¼ p
r . The first law becomes
U 1 À U 2 ¼ Q À
Z 2
1
pdV ¼ 0 À p
r V 2 À V 1
ð
Þ
i.e.,
U þ p
r V
ð
Þ 2 À U þ p
r V
ð
Þ 1 ¼ H 2 À H 1 ¼ 0
The entropy principle, Eq. (74), applies to this thermally isolated system, and we
have
S ¼ S
1
ð Þ H
1
ð Þ
; p
r
þ S
2
ð Þ H
2
ð Þ
; p
r
! S max
ð102BÞ
The equilibrium values of H
1
ð Þ
equi and H
2
ð Þ
equi are
160
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