4.3 Solutions
277
E S
E T
=
(∂ P/∂ V ) S
(∂ P/∂ V ) T
=
(∂ P/∂ V ) S
∂ T
∂ V
S
(∂ P/∂ S) T
∂ S
∂ V
T
=
(∂ T /∂ V ) S
∂ S
∂ P
T
(∂ T /∂ P) S
∂ S
∂ V
T
=
(∂ P/∂ S) V (∂ V /∂ T ) P
(∂ V /∂ S) P (∂ P/∂ T ) V
from the relations given in Problems 4.21 and 4.22
∴
E S
E T
=
(∂ S/∂ T ) P
(∂ S/∂ T ) V
=
(∂ Q/∂ T ) P
(∂ Q/∂ T ) V
=
C P
C V
= γ
4.37
(∂ V /∂ T ) S
(∂ V /∂ T ) P
=
1
(∂ T /∂ V ) S (∂ V /∂ T ) P
=
1
−
∂ P
∂ S
V
∂ V
∂ T
P
where we have used Eq. (23) of Problem 4.22.
Writing
∂ P
∂ S
V
=
∂ P
∂ T
V
∂ T
∂ S
V
=
(∂ P/∂ T ) V
(∂ S/∂ T ) V
(∂ V /∂ T ) S
(∂ V /∂ T ) P
=
(∂ S/∂ T ) V
(∂ P/∂ T ) V (∂ V /∂ T ) P
=
(∂ S/∂ T ) V
−(C P − C V )/T
(by Eq. (4.1) of Problem 4.27
T (∂ S/∂ T ) V
−(C P − C V )
=
C V
−(C P − C V )
=
1
1 − γ
4.38
(∂ P/∂ T ) S
(∂ P/∂ T ) V
=
1
(∂ T /∂ P) S (∂ P/∂ T ) V
=
1
∂ V
∂ S
P
∂ P
∂ T
V
=
1
∂ V
∂ T
P
∂ T
∂ S
P
∂ P
∂ T
V
=
(∂ S/∂ T ) P
∂ V
∂ T
P
(∂ P/∂ T ) V
=
T (∂ S/∂ T ) P
(C P − C V )
=
C P
(C P − C V )
=
γ
γ − 1
where we have used Eq. (4.24) of Problem 4.22 and the relation
C P = T
∂ S
∂ T
P